restrictions on sets of conjugacy class sizes in arithmetic progressions

We continue the investigation, that began in [m. bianchi, a. gillio and p. p. palfy, a note on nite groups in which the conjugacy class sizes form an arithmetic progression, ischia group theory 2010, world sci. publ., hackensack, nj (2012) 20{25.] and [m. bianchi, s. p. glasby and cheryl e. praeger, conjugacy class sizes in arithmetic progression, j. group theory, 23 no. 6 (2020) 1039{1056.], 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 cs(g). finite groups satisfying cs(g) = f1; 2; 4; 6g and f1; 2; 4; 6; 8g are classied in [m. bianchi, s. p. glasby and cheryl e. praeger, conjugacy class sizes in arithmetic progression, j. group theory, 23 no. 6 (2020) 1039{1056.] and [m. bianchi, a. gillio and p. p. palfy, a note on finite groups in which the conjugacy class sizes form an arithmetic progression, ischia group theory 2010, world sci. publ., hackensack, nj (2012) 20{25.], respectively, we demonstrate these examples are rather special by proving the following. there exists a finite group g such that cs(g) = f1; 2; 2+1; 23g if and only if = 1. furthermore, there exists a finite group g such that cs(g) = f1; 2; 2+1; 23; 2+2g and is odd if and only if = 1.