Let $G$ be a group and $Aut(G)$ denote the automorphisms group of $G$. Then we define the action $G$ on $Aut(G)$ given by $\alpha^{g}=\alpha^{{\phi_{g}}}=\phi_{g}^{-1}\circ\alpha\circ\phi_{g}$ and the action $Aut(G)$ on $G$ given by $g^{\alpha}=(g)\alpha$ for all $g \in G$, $\alpha\in Aut(G)$ and $\phi_{g} \in Inn(G)$.
Hence non abelian tensor product $G\otimes Aut(G)$ is defined.
We say a group G is $2_{\otimes}$-auto Engel group if $$[g, \alpha]\otimes\alpha = 1_{\otimes},$$ for all $g \in G$ and $\alpha\in Aut(G)$.
I want some non-trivial examples of $2_{\otimes}$-auto Engel groups.
Thank you
I don't think there are many possibilities. I would try to think of a group with $\mathbb{Z_{2}}$ center whose automorphism group is also $\mathbb{Z}_{2}$. Maybe some sort of semidirect product?