Possessive when abbreviated letter is plural

Nope, no apostrophe needed. HFK stands on its own as a pseudo-noun*. Really good question, though.

* The people who get upset at "ATM machine" and "PIN number" will disagree with me.

HFK's would also work, for the same reason.

what is a psuedo-noun, and why doesn't it require an apostrophe?

I'd think that to use no apostrophe would be if you were using it as an adjective, as to replace with the original phrase the possibilities are

"Help for Kids' activites will start in the summer."
"Help for Kids activites will start in the summer."

The first one shows possession as was intended, the second one merely shows description. When you replace with the acronym you get

"HFK' activites will start in the summer."

but the phrase no longer ends in 's' so it must be changed, for the same reason that you add an 's' when using the possessive form of plurals that don't end in 's' (that is, you wouldn't refer to "all the mice' cheese," you'd refer to "all the mice's cheese").

This makes the correct form

"HFK's activites will start in the summer."

All in my humble opinion, of course. I'm sure there's better arguments for this phrase then mine, but this is how I would do it.

than* mine, but this is how I would do it too.

If "HFK" is an accepted and frequently used abbreviation then it will be read as "ache eff kay" in which case "HFK's activities..." is the only right way to write this (if this expressing possession of the activities, not describing them, which would be "HFK activities..."). If HFK is not an accepted and frequently used abbreviation, then don't abbreviate at all. "HFK' activities... " is wrong in every case I can imagine (which isn't all that many).

I believe the correct spelling of the letter 'H' is aitch.

The "Kids" might be plural, but it is part of the initials for something that is singular.

"HFK's" to describe possession of the activities
"HFK" to describe the activities