Let $n$ be a positive integer, and $p$ a prime. Any subgroup $H\le \operatorname{GL}_n(\mathbb{F}_p)$ acts on the polynomial ring $\mathbb{F}_p[x_1,\ldots,x_n]$ via $A\cdot x_i=\sum_j a_{ji}x_j$ for all $A=(a_{ij})\in H$, and $A\cdot f(x_1,\ldots,x_n):=f(A\cdot x_1,\ldots,A\cdot x_n)$.

How can we compute the $\mathbb{F}_p$-algebra of invariants $\mathbb{F}_p[x_1,\ldots,x_n]^H$ given generators of $H$?

For example, take $H=\operatorname{SL}_n(\mathbb{F}_p)$. Then if we let

$$L_{n,s}=\left|\begin{array}{cccc} x_1&x_2&\cdots&x_n\\ x_1^{p}&x_2^{p}&\cdots&x_n^p\\ \cdots&\cdots&\cdots&\cdots\\ \widehat{x_1^{p^s}}&\widehat{x_2^{p^s}}&\cdots&\widehat{x_n^{p^s}}\\ \cdots&\cdots&\cdots&\cdots\\ x_1^{p^n}&x_2^{p^n}&\cdots&x_n^{p^n} \end{array}\right|$$

where the $p^s$ row is omitted, the invariants are generated by the Dickson polynomials $\dfrac{L_{n,s}}{L_{n,n}}$ for $s=0,\ldots,n-1$.

However, if $H$ is arbitrary (described by a list of generators $h_1,\ldots,h_m$), is there some algorithm for computing the invariants? Or perhaps at least an upper bound on the degree of the generators of the invariants so that we can be sure to have found all of them via some computer search?