Is it ok to use a registered trademark logo as a Facebook group picture?

Notice: The below is for educational and informational purposes only, is not a reflection on or a representation of any views or opinions held by my employer, and should not be construed as legal advice. Nothing herein is meant to create an attorney-client relationship.

Let's forget for the moment whether or not this might be considered a fair use. If the former employer politely asked you to remove the logo and the group photo includes not only former employees but current employees of this employer, as a practical matter, it might be worthwhile to agree to do so. This could save on potential legal fees in the event the employer pursues the matter and would most likely avoid creating some awkwardness with the current employees who appear in the group photo.

The former employer might find Facebook to be willing to get rid of the group if Facebook thought that it might have to defend against a trademark or copyright infringement case. Possible monetary damages are likely extremely small, but the cost of defending an action is more than zero, which is probably more than you want to spend as the administrator of a small Facebook group.
I'd just use a different picture.