Suppose we're working the category of modules over some ring $R$. Suppose a module $E$ is an extension of $M$ by $N$ in two different ways. In other words, I have two short exact sequences

<p>
\begin{array}{ccccccccc}
 0&\to &N&\xrightarrow{i_1}&E&\xrightarrow{p_1}&M&\to &0\\
  & & \wr\downarrow ?& & \wr\downarrow ?& & \wr\downarrow ?\\
 0&\to &N&\xrightarrow{i_2}&E&\xrightarrow{p_2}&M&\to &0
\end{array}
</p>

> Must there be an isomorphism between these two short exact sequences?

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##Alternative formulation

$Ext^1(M,N)$ parameterizes extensions of $M$ by $N$ modulo isomorphims *of extensions*. Suppose I'm interested in parameterizing extensions of $M$ by $N$ modulo *abstract isomorphisms* (which don't have to respect the submodule $N$ or the quotient $M$). One obvious thing to note is that there is a left action of $Aut(M)$ on $Ext^1(M,N)$, and that any two extensions related by this action are abstractly isomorphic. Similarly, there is a right action of $Aut(N)$ so that any two extensions related by the action are abstractly isomorphic.

> Does the quotient set $Aut(M)\backslash Ext^1(M,N)/Aut(N)$ parameterize extensions of $M$ by $N$ modulo abstract isomorphism?

Note: I'm **not** asking whether all abstract isomorphisms are generated by $Aut(M)$ and $Aut(N)$. They certainly aren't. I'm asking whether for every pair of abstractly isomorphic extensions *there exists* some isomorphism between them which is generated by $Aut(M)$ and $Aut(N)$.