Automorphism of two members as Generator

Question

Let $X=\langle a,b|a^{2^m}=b^{2^n}=1,[a,b]=a^{2^{m-1}}\rangle$, $m,n\ge 2$
If $\alpha \in Aut(X)$ (Automorphism Group of $X$) is defined as \begin{cases} \alpha(a)=a^{2^{m-1}+1}\\ \alpha(b)=b^{2^{n-1}+1} \end{cases} How we can show $X=\langle\alpha(a),\alpha(b)\rangle$?