  Motivated by recent problems regarding the symmetry of Hecke algebras, we
investigate the symmetry of the endomorphism algebra $E_P(M)$ for $P$ a
$p$-group and $M$ a $kP$-module with $k$ a field of characteristic $p$. We
provide a complete analysis for cyclic $p$-groups and the dihedral 2-groups.
For the dihedral 2-groups, this requires the classification of the
indecomposable modules in terms of string modules and band modules. We
generalize our techniques to consider $E_{\Lambda}(M)$ for $\Lambda$ a Nakayama
algebra, a local algebra, or even an arbitrary algebra.
