  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
$k:=T/M$, $\phi: T\to k$ the natural projection, and let $R={\phi}^{-1}(D)$.
For each star operation $\ast$ on $R$, we define the star operation $\ast_\phi$
on $D$, i.e., the ``projection'' of $\ast$ under $\phi$, 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, $0\to \Cl^{\ast_{\phi}}(D) \to \Cl^\ast(R) \to
\Cl^{{(\ast)}_{_{T}}}(T)\to 0$ is split exact. In particular, when $\ast =
t_{R}$, we deduce that the sequence $ 0\to \Cl^{t_{D}}(D) {\to} \Cl^{t_{R}}(R)
{\to}\Cl^{(t_{R})_{_{T}}}(T) \to 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.
