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We continue the investigation, that began in [3] and [4], into finite groupswhose 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 demonstratethese examples are rather special by proving the following. There exists afinite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1},2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finitegroup $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha +1}, 2^{\alpha}3,2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha=1$.

Restrictions on sets of conjugacy class sizes in arithmetic progressions