I'd guess that perhaps they *can* be supported on $G$ but that it is not always the case that they *must* be. Suppose that the entropy for both $X$ and $Y$ should be $1$. Then indeed if $n=2m$ we should (I'd think) take $X=Y=\lbrace0,m\rbrace$ with equal weighting and then get $X+Y=Z.$  But for $n$ odd (and composite) I'd think that $X$ and $Y$ should each be $\lbrace0,g\rbrace$ for some $g \ne 0$ with $X+Z$ supported on the set $\lbrace 0,g,2g \rbrace.$ with probabilities $1/4,1/4$ and $1/2.$ Of course we can also shift and have $X=\lbrace0,g\rbrace$ and $Y=\lbrace0,-g\rbrace.$ In any case, there is no gain (nor loss) in confining $g$ to a proper subgroup.