  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 ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1},
2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finite
group $G$ 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 $\alpha=1$.
