problem in understanding some part of proof of theorem

Question

In proof of theorem:
for every finite non abelian p-group $G$ of class 2 there is utter automorphism of calss 2 fixing $\Phi(G)$
by counterexample,If we know
1. $G'=\langle[a,b]\rangle$
2. $H=\langle a,b\rangle$
3. $G=H C_G(H)$
4. $Z(G)$ is cyclic
5. $exp(\frac{G}{Z(G)})=exp(\frac{H}{Z(H)})=2^n$ , ($exp(G')=max${$|[x,y]|:x,y\in G$})
How we can get $Z(H)=\langle a^{2^n},b^{2^n},[a,b]\rangle$
6. $n\ge2$
since $Z(H)$ is cyclic If we have $a^{2^ni}=b^{2^n}$ and i be even then
how we can show $a^{-2^ni}b^{2^n}=1\to (a^{-i}b)^{2^n}=1$? does this mean $a^{-i} and$ b are commutative?
7. $(a^{-i}b)^{2^{n-1}}\not\in Z(H)$
how we get $[a,b]=[a,a^{i}b]$ is of order $2^n$