Restrictions on sets of conjugacy class sizes in arithmetic progressions

We continue the investigation, that began in [3] and [4], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1,2,4,6\}$ and $\{1,2,4,6,8\}$ are classified in [4] and [3], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^α, 2^{α+1}, 2^α3 \}$ if and only if $α=1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^α, 2^{α+1}, 2^α3, 2^{α+2}\}$ and $α$ is odd if and only if $α=1$.

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