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Assuming you want that to hold for all $x,y \in G,$ any (non-Abelian) extra-special $2$-group $G$ will have that property. For there are only two squares in $G$: the identity, and the unique element of order $2$ in $Z(G).$ However, note that $(x^{p}y^{p})^{2}$ and $(y^{p}x^{p})^{2}$ are conjugate in $G$, so since they are both central, they are equal. By the way, it is easy to see that any finite group of odd order with the stated property is Abelian.