Let $A$ and $G$ be abelian varieties over $\mathbb{C}$. An element $P$ of $\text{Ext}(A, G)$ is an exact sequence

$0 \to G \to P \to A \to 0$,

here one can give $P$ the structure of an abelian variety. And $P$ can be viewed as principal $G$-bundle over $A$. In general, is there a way to determine if two given elements $P$ and $P'$ in $\text{Ext}(A, G)$ are isomorphic?

In particular, assume that $A$ is an elliptic curve. Given two extensions $0 \to G \to P \to A \to 0$ and $0 \to G \to P' \to A \to 0$ with morphisms $g : G \to G$, $f : P \to P'$, and $h : A \to A$ so that the resulting diagram commutes, $g$ is an isomorphism, and $h$ is an isogeny with kernel $(\mathbb{Z}/n\mathbb{Z})^2$. Can we show that $P$ and $P'$ are isomorphic or is there an example otherwise?