There are two nonabelian groups of order $p^3$, namely, semidirect product of $\mathbb Z /p \mathbb Z \times \mathbb Z /p \mathbb Z$ by $\mathbb Z /p \mathbb Z$ and semidirect product of $\mathbb Z /p^2 \mathbb Z$ by $\mathbb Z /p \mathbb Z$. What are the automorphism groups of these groups?

For the latter group, the answer is Bidwell & Curran's paper "The Automorphism Group of a Split Metacyclic pGroup" 


The former group can be seen as the group of unitriangular $3 \times 3$matrices over the field with $p$ elements: $$G = \left\{ \begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix} \right\} \leq SL(3,p)$$ The automorphism groups of such groups have been studied (in a much larger generality); see for instance the paper "The automorphism group of the group of unitriangular matrices over a field" by Ayan Mahalanobis (http://arxiv.org/abs/1012.5534v1). 


The automorphism groups of all pgroups of order p^3 can be found at http://www.math.kth.se/~boij/kandexjobbVT11/Material/pgroups.pdf 


There is a clear and more specific answer (with reference moreover!) here, despite the different question: http://math.stackexchange.com/a/18496/84625 In short $\operatorname{Aut}\left(\left(\mathbb{Z}_p \times \mathbb{Z}_p\right) \rtimes \mathbb{Z}_p) \right) \cong \operatorname{AGL}(2,p)$ while $\operatorname{Aut}\left(\mathbb{Z}_{p^2} \rtimes \mathbb{Z}_p \right) \cong \left(\mathbb{Z}_p \times \mathbb{Z}_p\right) \rtimes \operatorname{AGL}(1,p)$ where $\operatorname{AGL}(n,p)$ is the General Affine Group . 

