  Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite
simple unital C*-algebra, and let $\alpha \colon G \to \operatorname{Aut} (A)$
be a tracially strictly approximately inner action of $G$ on $A$. Then the
radius of comparison satisfies $\operatorname{rc} (A) \leq \operatorname{rc}
\big( C^*(G, A, \alpha) \big)$ and if $C^*(G, A, \alpha)$ is simple, then
$\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, \alpha) \big)
\leq \operatorname{rc} (A^{\alpha})$. Further, the inclusion of $A$ in $C^*(G,
A, \alpha)$ induces an isomorphism from the purely positive part of the Cuntz
semigroup $\operatorname{Cu} (A)$ to its image in $\operatorname{Cu}
\left(C^*(G, A, \alpha)\right)$. If $\alpha$ is strictly approximately inner,
then in fact $\operatorname{Cu} (A) \to \operatorname{Cu} \left(C^*(G, A,
\alpha) \right)$ is an ordered semigroup isomorphism onto its range. Also, for
every finite group $G$ and for every $\eta \in \left(0,
\frac{1}{\operatorname{card} (G)}\right)$, we construct a simple separable
unital AH algebra $A$ with stable rank one and a strictly approximately inner
action $\alpha \colon G \to \operatorname{Aut} (A)$ such that:
  (1) $\alpha$ is pointwise outer and doesn't have the weak tracial Rokhlin
property.
  (2) $\operatorname{rc} (A) =\operatorname{rc} \left(C^*(G, A, \alpha)\right)=
\eta$.
