It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly admit self-maps of degree $>1$. Moreover I guess if we take a compact Lie group and consider map $x\to x^2$ the degree of such map is $>1$ (in fact I did not check it).
Question. Let $G$ be a compact Lie group and $H$ its proper connected Lie subgroup. Is it true that $G/H$ admits self-maps of degree $>1$? If not, is there some classification of such quotients and that admit / don't admit a self-map with $deg>1$?