Let $k$ be a field (finite if you'd like), and let $f:A\to B$ be a map of graded, commutative $k$-algebras. Suppose further that $A$ is finitely generated and choose a presentation
$$A:=k[x_1,\ldots,x_n]/I$$
with $\deg{x_i}=d_i$. If $B'\subset B$ is a graded subalgebra, is there an algorithm for determining a presentation for $f^{-1}(B')\subset A$?
An idea: If you take $k$ to be finite, then for any nonnegative integer $m$, the set of homogeneous elements of degree $m$ in $A$, denoted $A_m$, is a finite set. So $f^{-1}(B')_m$ can be determined for any $m$ by applying $f$ to all $a\in A_m$ and determining whether or not $f(a)\in B'$. However, without some sort of bound on the degree of the generators of $f^{-1}(B')$, we wouldn't know when to stop computing $f^{-1}(B')_m$. Even if we knew when to stop, it is not clear to me how we could determine generators and relations from the graded pieces.
I've been unable to find such an algorithm implemented in Magma. Perhaps another CAS has one?