Let
$$1 \to G \overset{\iota_i}{\to} H_i \overset{\pi_i}{\to} K \to 1 $$ be two central extensions of $K$ by $G$. ($i=1,2$). Assume for simplicity that this is an extension of Lie groups, and $G$ is closed, so $\pi_i: H_i \to K$ is a pointed principal $G$-bundle over $K$.
If $\pi_i : H_i \to K$ are isomorphic as bundles (i.e. there is a $G$-equivariant map $f: H_1 \to H_2$), are the two extensions also isomorphic (as extensions)?