On the star class group of a pullback

For the domain $R$ arising from the construction $T, M, D$, we relate the star class groups of $R$ to those of $T$ and $D$. More precisely, let $T$ be an integral domain, $M$ a nonzero maximal ideal of $T$, $D$ a proper subring of $k:=T/M$, $\varphi: T\rightarrow k$ the natural projection, and let $R={\varphi}^{-1}(D)$. For each star operation $\ast$ on $R$, we define the star operation $\ast_\varphi$ on $D$, i.e., the ``projection'' of $\ast$ under $\varphi$, and the star operation ${(\ast)}_{_{\!T}}$ on $T$, i.e., the ``extension'' of $\ast$ to $T$. Then we show that, under a mild hypothesis on the group of units of $T$, if $\ast$ is a star operation of finite type, then the sequence of canonical homomorphisms $0\rightarrow \Cl^{\ast_{\varphi}}(D) \rightarrow \Cl^\ast(R) \rightarrow \Cl^{{(\ast)}_{_{\!T}}}(T)\rightarrow 0$ is split exact. In particular, when $\ast = t_{R}$, we deduce that the sequence $ 0\rightarrow \Cl^{t_{D}}(D) {\rightarrow} \Cl^{t_{R}}(R) {\rightarrow}\Cl^{(t_{R})_{_{\!T}}}(T) \rightarrow 0 $ is split exact. The relation between ${(t_{R})_{_{\!T}}}$ and $t_{T}$ (and between $\Cl^{(t_{R})_{_{\!T}}}(T)$ and $\Cl^{t_{T}}(T)$) is also investigated.