Another character theoretic formula for base size

A base for a permutation group G acting on a set $\Omega$ is a sequence $B$ of points of $\Omega$ such that the pointwise stabiliser $G_B$ is trivial. The base size of G is the size of a smallest base for G. Extending the results of a recent paper of the author, we prove a 2013 conjecture of Fritzsche, Külshammer, and Reiche. Moreover, we generalise this conjecture and derive an alternative character theoretic formula for the base size of a certain class of permutation groups. As a consequence of our work, a third formula for the base size of the symmetric group of degree n acting on the subsets of $\{1,2,\cdots ,n\}$ is obtained.