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A digraph is associated with a finite group by utilizing the power map $f:G\to G$ defined by $f\left(x\right)=x^k$ for all $x\in G$, where $k$ is a fixed natural number. It is denoted by ${\gamma }_G(n,k)$. In this paper, the generalized quaternion and $2$-groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a $2$-group are determined for a $2$-group to be a generalized quaternion group. Further, the classification of two generated $2$-groups as abelian or non-abelian...