  Let $G$ be a finite group and $\cF$ be a family of subgroups of $G$ closed
under conjugation and taking subgroups. We consider the question whether there
exists a periodic relative $\cF$-projective resolution for $\bbZ$ when $\cF$ is
the family of all subgroups $H \leq G$ with $\rk H \leq \rk G-1$. We answer
this question negatively by calculating the relative group cohomology $\cF H^*
(G, \bbF_2)$ where $G=\bbZ/2\times \bbZ /2$ and $\cF$ is the family of cyclic
subgroups of $G$. To do this calculation we first observe that the relative
group cohomology $\cF H^*(G, M)$ can be calculated using the ext-groups over
the orbit category of $G$ restricted to the family $\cF$. In second part of the
paper, we discuss the construction of a spectral sequence that converges to the
cohomology of a group $G$ and whose horizontal line at $E_2$ page is isomorphic
to the relative group cohomology of $G$.
