A generalized character related to the local structure and representation theory of a finite group

We consider the generalized character ${\mathrm{\Psi }}_{1,p,G}$ of a finite group $G$ which vanishes on all $p$-singular elements of $G$ and whose value at each $p$-regular $y\in G$ is the number of $p$-elements of $C_G(y)$. We conjecture that this is always a character, and may be afforded by a projective $RG$-module, where $R$ is an appropriate complete discrete valuation ring whose residue field has characteristic $p$. We examine a number of case where this is the case, and consider consequences for the representation theory and character theory of $G$ when this conjecture is known to hold. In particular, we prove, among other things, that the conjecture is valid for all primes $p$ in the case that $G\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,q)$ or $\mathrm{S}\mathrm{L}(2,q)$ for every prime power $q$.