Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions: $$ B(B^\intercal AB)^\star B^\intercal = A^\star, $$ and such that $A^\star=0$ if and only if $A=0$?

Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) \to \operatorname{M}(n,m,\mathbb R)$.

Edit: If possible, we would also like $\operatorname{rank}(A^\star)=\operatorname{rank}(A)$.