  When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$),
we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When,
however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital
C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit
such that $D$ has \PSP and $\sup_{p\in P(D)}\tsr(pAp) < \infty$, then $\tsr(A)
\leq 2.$ As an application let $A$ be a simple unital \ca with $\tsr(A) = 1$
and \PSP, $\{G_k\}_{k=1}^n$ finite groups, $\af_k$ actions from $G_k$ to ${\rm
Aut}((...((A\times_{\af_1}G_1)\times_{\af_2}
G_2)...)\times_{\af_{k-1}}G_{k-1}).$ $(G_0 = \{1\})$ Then $$ \tsr((...
((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_n}G_n) \leq 2. $$
