A module M R is called principally quasi-Baer (or simply p.q.-Baer) if the annihilator of every cyclic submodule of M R is generated by an idempotent, as a right ideal. Let α be an automorphism of R and M R be an α-compatible module and every countable subset of right semicentral idempotents in R has a generalized countable join or R satisfies the ACC on left annihilator ideals. It is shown that M R is p.q.-Baer if and only if M[[x]] R[[x; α]] is p.q.-Baer if and only if M[[x, x −1]] R[[x, x −1; α]] is p.q.-Baer. As a consequence, we unify and extend nontrivially many of the previously known results such as [11, 15, 20]. Examples to illustrate and delimit the theory are provided.