Lie powers of the natural module for GL(2,K)

In recent work of R. M. Bryant and the second author a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural $\rm{GL}(n,K)$ was presented. There is was shown that nearly all of the indecomposable summands of the $r$th tensor power also occur up to isomorphism as summands of the $r$th Lie power provided that $r\neq p^m$ and $r \neq 2p^m$, where $p$ is the characteristic of $K$. In the current paper we restrict attention to ${\rm GL}(2,K)$ and consider the missing cases where $r = p^m$ and $r = 2p^m$. In particular, we prove that the indecomposable summand of the $r$th tensor power of the natural module with highest weight $(r-1,1)$ is a summand of the $r$th Lie power if and only if $r$ is a not power of $p$.