Short Five Lemma for Nonabelian Groups? Is there a version of the "short five lemma" that applies to nonabelian groups? Perhaps it would incorporate some extra information about splittings. I have in mind the following kind of statement:
Theorem: Let $K,K',G,G',H,H'$ be groups and consider the following commutative diagram where the top and bottom rows are exact:
$$\require{AMScd}\begin{CD}
 1 @>>> K @>>> G @>>> H @>>> 1\\
{} @V{\alpha}VV @V{\beta}VV @V{\gamma}VV {} \\
1 @>>> K' @>>> G' @>>> H' @>>> 1
\end{CD}$$
Suppose that we also have maps $s:H\to G$ and $s':H'\to G'$ satisfying (blah). If $\alpha$ and $\gamma$ are isomorphisms satisfying (blah) then it follows that $\beta$ is an isomorphism satisfying (blah). ///
Any idea what blah should be?
 A: Let us write $p: G\to H$, $p':G'\to H'$, $k:K\to G$ and $k':K'\to G'$ for the unlabeled arrrows in your diagram.
Suppose $g$ is in the kernel of $\beta$ (so that $\beta(g)=1$), we will show that $g=1$. We have $1=p'(\beta(g))=\gamma (p(g))$ and so $p(g)=1$ since $\gamma$ is an isomorphism. Since the top row is exact it follows that there is $x$ in $K$ such that $k(x)=g$. We have $k'(\alpha(x)) = \beta(k(x))=\beta(g)=1$ and hence $x=1$ since $\alpha$ is an isomorphism and $k'$ is a monomorphism. Therefore $g=k(1)=1$ and so $\beta$ is a monomorphism.
Now suppose that $g'$ is an element in $G'$ we will show that there is an element in $G$ which gets mapped by $\beta$ to $g'$. Since $\gamma$ is an isomorphism and $p$ is surjective it follows that there is $g_1$ in $G$ such that $\gamma(p(g_1)) = p'(g')$. It follows that $p'(g' \beta(g_1)^{-1})=p'(g')p'(\beta(g_1))^{-1}=p'(g')\gamma(p(g_1))^{-1} =p'(g')p'(g')^{-1}=1$ and hence, since the bottom row is exact, there is $x'$ in $X'$ such that $k'(x')=g'\beta(g_1)^{-1}$. But since $\alpha$ is an isomorphism, there is $x$ in $X$ such that $\alpha(x)=x'$. We have $\beta(k(x)g_1) = \beta(k(x))\beta(g_1)=k'(\alpha(x))\beta(g_1)=k'(x)\beta(g_1)=g'\beta(g)^{-1}\beta(g)=g'$. This proves that $\gamma$ is surjective.
In fact more is true:
(a) If $\alpha$ and $\gamma$ are monomorphisms, then so is $\beta$.
(b) If $\alpha$ and $\gamma$ are surjective, then so is $\beta$.
At a slighty higher level one can prove that for any variety of universal algebras which have only one constant $e$. The short five lemma holds if and only if for some naturnal number $n$ there are $n$ binary terms $s_i(x,y)$ and one $n+1$-ary term $p(x_1,...,x_{n+1})$ such that $s_i(x,x)=e$ $p(s_1(x,y),s_2(x,y),...,s_n(x,y),y)=x$. For groups $n=1$, $s_1(x,y)=xy^{-1}$ and $p(x,y)=xy$.
