You are even closer now!
What the type checker is trying to tell you is that it doesn't know how to raise Maybe Integer to some power.
You apply the ^ operator to two values, f2Maybe (n `div` 2) and 2. Let us give them names:
let a = f2Maybe (n `div` 2) :: Maybe Integer
b = 2 :: Int
in a ^ b
You see that the type of a is Maybe Integer. What does this mean? There are only 2 cases to consider. You have Just an integer or you have Nothing.
You can use the case construct to write the code for both cases.
f2Maybe :: Integer -> Maybe Integer
f2Maybe n
| n > 0 = Nothing
| n == 0 = Just 1
| even n = case f2Maybe (n `div` 2) of
Just x -> <fill in>
Nothing -> <fill in>
| odd n = case f2Maybe (n `div` 2) of
Just x -> <fill in>
Nothing -> <fill in>