possessives - "Whomever runs it's" or "whomever runs its"?

I know that "its" is the possessive form of "it", but does this rule apply to the possessive form of phrases ending in "it"? Should I say, "the program runs on whomever runs its computer" or "the program runs on whomever runs it's computer" (also I hope I'm right that that should be "whomever" and not "whoever")?

Answer

NOTE: Ignore this first bit and skip down to the EDIT section for the right answer. I misread the sentence on first (and second, and third) reading.

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The closest sentence with correct grammar (but not sense) that matches your own is:

The program runs on whoever runs its computer.

Because the object of the preposition on is not **whomever*, which is wrong, but rather the entire clause whoever runs its computer, in which whoever acts as the subject of the verb runs. The proximity of on is a red herring — or strange attractor, depending on which metaphor you prefer.

Since you can replace its with his — and not with it is — this shows that its must be correct.

However, all that said and done, I am unconvinced that this sentence makes sense. I bet you want:

The program runs on the computer of whoever runs it.

Because I don’t think it runs on people, but rather on computers.

Maybe you could almost get away with:

The program runs on whosever computer runs it.

But that seems a bit off, too.

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EDIT

Now I see what you were going for! You were trying to use the Saxon genitive on the entire noun phrase:

The program runs on (whoever runs it)’s computer.

Which is just like:

The program runs on (the person who runs it)’s computer.

That way it would work the same way as these do:

It’s (the Queen of England)’s castle.
It’s (the man at the door)’s hat.
It’s (my wife and I)’s dinner.
He’s (all of those kids)’ friend.
He’s (all of us)’s friend.

Unfortunately, without parentheses, people will too easily misread it, as in fact did I the first few times through it.

While there is certainly some sense in what you say, it if nothing else garden-paths to the wrong it’s, which makes it look like a mistake even when it isn’t.

I’d still say it runs on the computer of whoever runs it myself, just to make sure.