Yes, but such low-information situations are fairly rare.

One particularly controversial such case is police stop-and-frisks.

This is possibly a case where group means actually are relevant, yes, modulo a lot of assumptions about how people are selected for the stop-and-frisk.

Or take racial profiling in airports:

Again, not objectionable on the surface, However, given the stunningly ineffective nature of airport security (cf. Bruce Schneier and the "security theater" concept) I doubt this actually provides a benefit, for reasons wholly unrelated to race.

As an aside, in adversarial situations you need to be careful that weighted targetting based on superficial cues doesn't merely give the enemy information on how to disproportionately avoid scrutiny.

EDIT: Either I missed this part or you added it while I was replying, but:

Would you or wouldn't you allow unsupervised immigration from a certain country based on the average IQ there?

Assuming you want to filter applicants based on IQ, testing individuals seems vastly more helpful than assuming based on population mean, especially given that the demographics of applicants will not be the same as the total population. Also, if you admit all immigrants from any country you're likely to get a self-selected group of people who are less successful at home.