Let $G$ be a finite group, let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, and let $α\colon G \to \operatorname{Aut} (A)$ be a tracially strictly approximately inner action of $G$ on $A$. Then the radius of comparison satisfies $\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big)$ and if $C^*(G, A, α)$ is simple, then $\operatorname{rc} (A) \leq \operatorname{rc} \big( C^*(G, A, α) \big) \leq \operatorname{rc} (A^α)$. Further, the inclusion of $A$ in $C^*(G, A, α)$ induces an isomorphism from the purely positive part of the Cuntz semigroup $\operatorname{Cu} (A)$ to its image in $\operatorname{Cu} \left(C^*(G, A, α)\right)$. If $α$ is strictly approximately inner, then in fact $\operatorname{Cu} (A) \to \operatorname{Cu} \left(C^*(G, A, α) \right)$ is an ordered semigroup isomorphism onto its range. Also, for every finite group $G$ and for every $η\in \left(0, \frac{1}{\operatorname{card} (G)}\right)$, we construct a simple separable unital AH algebra $A$ with stable rank one and a strictly approximately inner action $α\colon G \to \operatorname{Aut} (A)$ such that: (1) $α$ is pointwise outer and doesn't have the weak tracial Rokhlin property. (2) $\operatorname{rc} (A) =\operatorname{rc} \left(C^*(G, A, α)\right)= η$. 35 Pages