Let $G$ be a finite Abelian group with endomorphism ring $End(G)$. I am interested in the probability $P(\phi(g_1) = g_2)$ for fixed $g_1,g_2 \in G$ and a uniformly chosen endomorphism $\phi(\cdot)$ from $End(G)$. Essentially, I want to understand where the set of endomorphisms will take each element $g \in G$. I ran into this question while considering homomorphic compression schemes that compress an $n$-length sequence $g^n$ into a sequence of length $k$ by applying a homomorphism $\phi \colon G^n \rightarrow G^k$. I describe the question in detail below.

Let $\mathbb{Z}_n$ be the cyclic group of $n$ elements. If $G ={\mathbb{Z}_{p^r}}$, I understand what is going on and can prove for instance that $\phi(g)$ is uniformly distributed across the smallest subgroup of $\mathbb{Z}_{p^r}$ that $g$ belongs to as $\phi(\cdot)$ varies over $End(\mathbb{Z}_{p^r})$. But, I am having trouble understanding what happens in the case of groups of the form $\mathbb{Z}_{p^r}^k$ such as $\mathbb{Z}_2^2$ for example. In this case, $\phi(g)$ is uniformly distributed over $\mathbb{Z}_2^2$ for all non-identity $g$ regardless of which subgroup $g$ belongs to.

Question: Is there a uniform way to write down the probability $P(\phi(g_1) = g_2)$ for fixed $g_1,g_2 \in G$ and an arbitrary $\phi(\cdot) \in End(G)$ for a finite Abelian group $G$?

I would greatly appreciate any pointers and hope the question isn't too elementary for MO. Please feel free to edit/re-tag the question if needed.