A cone is a $R_+$-module. That is, a cone is an abelian monoid that is closed under nonnegative real scalar multiplication. An automorphism of a cone is a bijective $R_+$-linear map. That is a map $f:C\to C$ such that $f(\alpha x +\beta y)=\alpha f(x) + \beta f(y)$ for all $\alpha,\beta \ge 0 $ and $x,y\in C$.
Suppose I have an inclusion of cones $C\subset D$. If the automorphism group $Aut(C)$ and $Aut(D)$ are isomorphic, does that imply that $C=D$? You may assume that these cones are finite-dimensional, in the sense that there is an $R_+$-linear map from each of these cones to some euclidean space $R^d$.