Projective class group of crossed product

Let $R$ be a Dedekind ring, ${C_0}(R[\pi ,\rho ])$ the projective class group of the crossed product $R[\pi,\rho]$, $M$ some class of subgroups of a finite group $\pi$, $C_0^M(R[\pi ,\rho ])$ the sum of the images of the maps ${C_0}(R[\pi',\rho])\to{C_0}(R[\pi,\rho])$ induced by $i:\pi'\subset \pi$, $\pi ' \in M$. The main result of the paper is the study of some properties of the exponents of $C_0^M(R[\pi ,\rho ])$ in $C(R[\pi ,\rho ])$ for cyclic, elementary, and hyperelementary subgroups of $\pi$, under natural restrictions on $R$ and $\pi$.