By short-sequences the exercise means sequences where $Im \ f \subset Ker \ g$.
These $H_n$ are homologies, analogous to this construction for groups, but here it's for modules. Normally I'd show my thinking, but for this exercise I don't even know where to start. Obviously I must show that $\alpha_{*}\circ \partial = \partial'\circ \gamma$, but what is this $_*$?
I'd be grateful for some clarification and I antecipate that I'm opening a bounty to reward an answer for this one. Thank you.
Sorry but I can't comment yet, so I'll put a brief answer.
First the $_*$ is pushforward / direct image of algebraic cycles. What you are asked to do is prove commutativity of the second diagram. Have you heard of / constructed the "long exact sequence" induced by a short exact sequence yet? That may help.