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I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Flags with a horizontal black stripe in the design: ANGOLA, KENYA, EGYPT, BOTSWANA. (Prompted by @JeremyDover in comments)

4. Countries with a female leader: FINLAND (Sanna Marin, PM), DENMARK (Mette Frederiksen, PM), GABON (Rose Raponda, PM), BARBADOS (Mia Mottley, PM).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) it includes a black horizontal stripe in its design, and (iv) the current (female) Chancellor of Germany is Angela Merkel.

_Note:

_{I suspect the prominent mention of "right now (and that’s important to remember!)" in the flavour text is an indication that something related to current affairs (and which may change in the future) is key to the puzzle. With the fourth category focused on politics (where leaders may change in elections) I think this hidden hint is satisfied.}

I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Flags with a horizontal black stripe in the design: ANGOLA, KENYA, EGYPT, BOTSWANA. (Prompted by @JeremyDover in comments)

4. Countries with a female leader: FINLAND (Sanna Marin, PM), DENMARK (Mette Frederiksen, PM), GABON (Rose Raponda, PM), BARBADOS (Mia Mottley, PM).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) it includes a black horizontal stripe in its design, and (iv) the current (female) Chancellor of Germany is Angela Merkel.

_Notes:

<sub>I suspect the prominent mention of "right now (and that’s important to remember!)" in the flavour text is an indication that something related to current affairs (and which may change in the future) is key to the puzzle. With the fourth category focused on politics (where leaders may change in elections) I think this hidden hint is satisfied.

Also, there is a hint concealed in the title - the initials of 'Don't Embarrass Us' are 'DEU', the common 3-letter abbreviation for 'Deutschland' - the country's name in German!</sub>

I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Flags with a horizontal black stripe in the design: ANGOLA, KENYA, EGYPT, BOTSWANA. (Prompted by @JeremyDover in comments)

4. Countries with a female leader: FINLAND (Sanna Marin, PM), DENMARK (Mette Frederiksen, PM), GABON (Rose Raponda, PM), BARBADOS (Mia Mottley, PM).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) it includes a black horizontal stripe in its design, and (iv) the current (female) Chancellor of Germany is Angela Merkel.

I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Flags with a horizontal black stripe in the design: ANGOLA, KENYA, EGYPT, BOTSWANA. (Prompted by @JeremyDover in comments)

4. Countries with a female leader: FINLAND (Sanna Marin, PM), DENMARK (Mette Frederiksen, PM), GABON (Rose Raponda, PM), BARBADOS (Mia Mottley, PM).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) it includes a black horizontal stripe in its design, and (iv) the current (female) Chancellor of Germany is Angela Merkel.

_Note:

_{I suspect the prominent mention of "right now (and that’s important to remember!)" in the flavour text is an indication that something related to current affairs (and which may change in the future) is key to the puzzle. With the fourth category focused on politics (where leaders may change in elections) I think this hidden hint is satisfied.}

I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Places whose names contain a 4-letter word in English: BOTSWANA, BARBADOS, FINLAND, DENMARK.

4. Flags with horizontal stripes in the design: ANGOLA (2), GABON (3), KENYA (5), EGYPT (3).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) GERMANY contains both 'GERM' and 'MANY', and (iv) it includes 3 horizontal stripes in its design.

I think the missing flag is:

i.e. the flag of GERMANY

The flags depicted are (left to right, top to bottom):

ANGOLA, OMAN, MANITOBA, FINLAND;
KENYA, ROMANIA, DENMARK, FRANCE;
POLAND, CAYMAN ISLANDS, GABON, EGYPT;
JAPAN, BARBADOS, BOTSWANA, TENNESSEE.

The four connecting groups are:

1. Places containing 'MAN': OMAN, MANITOBA, ROMANIA, CAYMAN ISLANDS.

2. Places which have chemical elements named after them: FRANCE (Francium and Gallium), POLAND (Polonium), JAPAN (Nihonium), TENNESSEE (Tennessine).

3. Flags with a horizontal black stripe in the design: ANGOLA, KENYA, EGYPT, BOTSWANA. (Prompted by @JeremyDover in comments)

4. Countries with a female leader: FINLAND (Sanna Marin, PM), DENMARK (Mette Frederiksen, PM), GABON (Rose Raponda, PM), BARBADOS (Mia Mottley, PM).

A flag which therefore satisfies all 4 of these groups is:

Germany's, since (i) GERMANY contains 'MAN', (ii) the element Germanium is named after Germany, (iii) it includes a black horizontal stripe in its design, and (iv) the current (female) Chancellor of Germany is Angela Merkel.