Let $G = \prod_p \mathbb{Z}/p\mathbb{Z}$, where $p$ ranges over all primes, considered as an abelian group. What does $\text{Aut}(G)$ (or even $\text{End}(G)$) look like?

I know that that if we take $t(G) = \oplus_p \mathbb{Z}/p\mathbb{Z}$, then $\text{Aut}(t(G)) = \prod_p (\mathbb{Z}/p\mathbb{Z})^\times$ (can be thought of as infinite diagonal matrices), but I am more interested in the product.

More generally, if $G = \prod_{i \in I} G_i$, where each $G_i$ is fully invariant (embedded in the obvious way), what can we say about $\text{Aut}(G)$?

EDIT: I do apologize for the imprecision of the question, but anything would be helpful, and an exact description, as in the case for the direct sum, would be fantastic.