A simultaneous variant of the Hall-Paige theorem and applications

For $k\geq 1$ , we say that a finite group $G$ satisfies property $\mathcal P_k$ if, for every $β_1,\ldots,β_k\in\operatorname{Aut}(G)$, there exists a bijection $f$ of $G$ such that each map $g\mapsto β_i(g)f(g)$ is a bijection. For $k=1$ this is equivalent to the classical Hall--Paige complete map