automorphisms of a finite $p$-group If $G=\langle a, b : a^{p^{n}}= 1= b^{p^{n+1}}, [b, a]= b^{p} \rangle$, such that 
 $n\geq 2$ and $p$ is an odd prime number, then how can I define a non-inner automorphism of $G$? Is it possible to find the structure of all automorphisms of $G$?
 A: I'm assuming $[x,y]=x^{-1}y^{-1}xy$, but I expect that using the other convention you would get similar results.
Note that $a^{-1}ba = b^{1+p}$, and hence $a^{-1}b^pa = b^{p+p^2}\in\langle b^p\rangle$. Thus, $\langle b^p\rangle\triangleleft G$, and $G/\langle b^p\rangle$ is abelian, so $G$ is $2$-generated and has cyclic commutator subgroup.
For odd prime, such groups were studied by Miech: 
Miech R.J. On $\mathbf{p}$-groups with a cyclic commutator subgroup.
J. Austral. Math. Soc. 20 (1975), no. 2, 178–198. MR0404441 (53 #8243)
Miech describes every such group in terms of a 12-tuple of integers satisfying certain conditions. Namely:

Theorem 1. Let $p$ be an odd prime and let $G$ be a finite nonabelian $p$-group generated by two elements such that the commutator subgroup of $G$ is cyclic. Then $G$ has a pair of generators $\{x,y\}$ such that the defining relations of the group are:
  $$\begin{alignat*}{3}
[y,x]&= z, &\quad x^{p^{\alpha}} &= z^{Rp^{\rho}}, &\quad y^{p^{\beta}} &= z^{Sp^{\sigma}} \\
z^{p^{\gamma}} &= 1, &\quad [z,x] &= z^{Mp^{\mu}}, &\quad [z,y] &= z^{Np^{\nu}}
\end{alignat*}$$
  where $\alpha,\beta,\gamma,\rho,\sigma,\mu,\nu$ and $R,S,M,N$ are integers that satisfy the conditions
  $$ \alpha\geq \beta, \quad \alpha\geq \gamma,\quad \rho+\mu\geq \gamma,\quad \rho+\nu\geq \gamma,\quad \sigma+\nu\geq \gamma,\\
1\leq \mu,\quad \nu\leq\gamma,\quad 0\leq \rho,\quad \sigma\leq\gamma,\quad p^{\beta} - MSp^{\mu+\sigma}\equiv 0\pmod{p^{\gamma}}.$$
  Conversely, given any set of parameters $\{\alpha,\beta,\gamma,\rho,\sigma,\mu,\nu,R,S,M,N\}$ that satisfies these conditions there is a group defined by these relations.

Your groups would then be given by $\alpha=n$, $\beta=1$, $\gamma=n$, $\rho=n$, $\sigma=0$, $\mu=1$, $\nu=n$, and $R=S=M=N=1$. 
If you take $\alpha,\beta,\gamma$ as fixed, the groups can be succinctly described then by a $4$-tuple, $[Rp^{\rho}, Sp^{\sigma}, Mp^{\mu}, Np^{\nu}]$.
Section 1 of the paper describes the isomorphisms/automorphisms in Theorem 9 in terms of four congruences (a bit hard to type them here, since I've had to change some of the notation to avoid using $a$ and $b$, which you use in your description). This should allow you to count the automorphisms and figure out which ones are given by conjugation, though it will probably take some number crunching.
