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Dedicated to N. Christopher Phillips on the occasion of his 64th birthday.

Introduction {#Sec_Intro}
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Comparison theory of projections plays an important role in the type classification of factors. A C\*-algebra might have few or no projections. In this case comparison theory of projections may say nothing about the structure of the C\*-algebra. The appropriate substitute for projections is positive elements. This idea was first introduced by Cuntz in [@Cun78] for the purpose of studying dimension functions on simple C\*-algebras. Later, the radius of comparison of C\*-algebras, based on the Cuntz semigroup, was introduced by Andrew S. Toms in Section 6 of [@Tom06] to study exotic examples of simple amenable C\*-algebras that are not $\mathcal{Z}$-absorbing. In the commutative setting, it is well known that the radius of comparison of $C(X)$ is bounded above by one half the covering dimension of $X$ [@BRTTW12; @EN13]. Also, the comparison theory can be viewed as a non-commutative dimension theory [@AA20]. Remarkable progress has been achieved on the comparison theory in [@BRTTW12; @HP19; @Ni14; @Ph16; @Ph14]. In this paper, we consider the relation between comparison theory in a simple C\*-algebra $A$ and in the crossed product of $A$ by a finite group, under a 