In this paper we compute the four-loop corrections to the QED photon
self-energy $\Pi(Q^2)$ in the two limits of $q = 0$ and $Q^2 \to
\infty$. These results are used to explicitly construct the conversion
relations between the QED charge renormalized in on-shell(OS) and $\MSbar$
scheme. Using these relations and results of Baikov et al. \cite{Baikov:2012zm} we
construct the momentum dependent part of $\Pi(Q^2,m,\alpha)$ at
large $Q^2$ at five loops in both $\MSbar$ and OS schemes. As a direct
consequence we arrive at the full result for the QED $\beta$-function
in the OS scheme at five loops. These results are applied, in turn, to
analytically evaluate a class of asymptotic contributions to the muon
anomaly at five and six loops.