Invariant class sizes and solvability of finite groups under coprime action

Suppose that A and G are finite groups of relatively prime orders such that A acts on G via automorphisms, and then, A acts on the set of conjugacy classes of G. We study several arithmetical properties on the sizes of certain classes which are left fixed by A and how they reflect on the A-invariant structure and may imply the solvability of G.The first author is supported by the Valencian Government, Proyecto PROMETEOII/2015/011 and by Universitat Jaume I, grant P11B2012-05. The second author is supported by the research Project NNSF of China (Grant No. 11301218), University of Jinan Research Funds for Doctors (XBS1335 and XBS1336), and the Nature Science Fund of Shandong Province (No. inline imageR2014AM020)