On the category of modules of second kind for Galois coverings

Let F: R → R/G be a Galois covering and ${\displaystyle {mod}_1\left(\frac{R}{G}\right)}$ (resp. ${\displaystyle {mod}_2\left(\frac{R}{G}\right)}$) be a full subcategory of the module category mod (R/G), consisting of all R/G-modules of first (resp. second) kind with respect to F. The structure of the categories ${\displaystyle \frac{mod\left(\frac{R}{G}\right)}{{mod}_1\left(\frac{R}{G}\right)}}$ and ${\displaystyle {mod}_2\left(\frac{R}{G}\right)}$ is given in terms of representation categories of stabilizers of weakly-G-periodic modules for some class of coverings.