A character theoretic formula for base size II

A base for a permutation group $G$ acting on a set $\mathrm{\Omega }$ is a sequence $\mathcal{B}$ of points of $\mathrm{\Omega }$ such that the pointwise stabiliser $G_{\mathcal{B}}$ 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 $\{1,2,\dots ,n\}$ is obtained.