The interest of this paper is a characterization of graphs that induce periodic Grover walks with given periods. In previous studies, Yoshie has shown that the only graphs that induce odd periodic Grover walks are cycle graphs. However, this problem is largely unsolved for even periods. In this study, we show that regular graphs that induce $2l$-periodic Grover walks are also cycle graphs in most cases, where $l$ is an odd integer. The proof uses Galois theory.