Here is a problem from my Graduate Abstract Algebra course. I'm not quite sure how to go about part d at all, though the rest of the parts were easily proved using some basic machinery I already knew. I tried constructing a chain of subgroups that eventually formed a Galois Connection, but my TA noted that there was a small error in my proof that essentially nullified the result. Any help is useful and a full solution of part (d) would greatly be appreciated.
Hint: You need to show that $\phi(S)\subseteq T$ if and only if $\phi^{-1}(T)\subseteq S$ for subgroups $S\subseteq G$ and $T\subseteq H$.