Given a group G of k-linear automorphisms of a locally bounded k-category R it is proved that the endomorphism algebra $En{d}_R\left(B\right)$ of a G-atom B is a local semiprimary ring (Theorem 2.9); consequently, the injective $En{d}_R\left(B\right)$-module ${\left(En{d}_R\left(B\right)\right)}^{*}$ is indecomposable (Corollary 3.1) and the socle of the tensor product functor $-{\otimes }_R{B}^{*}$ is simple (Theorem 4.4). The problem when the Galois covering reduction to stabilizers with respect to a set U of periodic G-atoms (defined by the functors ${\Phi }^U:{\coprod }_{B\in U}modk{G}_B\to mod(R/G)$ and ${\Psi }^U:mod(R/G)\to {\prod }_{B\in U}modk{G}_B$)is full (resp. strictly full) is studied...