If $H = S_n$ then then the fundamental symmetric polynomials allow to write any $S_n$-invariant polynomial $f$ as a polynomial expression of these elementary symmetric functions. In other words, $\mathbb{C}[x_1, \dots ,x_n]^{S_n} = \mathbb{C}[e_1, \dots ,e_n]$.
Now, given $H \le S_n$ a subgroup of the symmetric group, is there a general way to compute a system of invariants for $\mathbb{C}[x_1, \dots , x_n]^H$ ?
EDIT:
A brute-force approach to find invariants for $\mathbb{C}[x_1, \dots,x_n]^H$ might be the Reynolds Operator?
It is defined as:
$$R : \mathbb{R}[x_1, \dots, x_n] \rightarrow \mathbb{R}[x_1, \dots, x_n]^H$$
$R(f):= \frac{1}{|G|}\sum_{g \in G}f(g \cdot \textbf{x})$$R(f):= \frac{1}{|H|}\sum_{g \in H}f(g \cdot \textbf{x})$.
So just for simplicity consider $\mathbb{Z}_3 \le S_3$ and we define a group action
\begin{align} \mathbb{Z}_3 \times V &\rightarrow V\\ (g, \textbf{x}) &\mapsto g \cdot \textbf{x} := (x_1+g, \dots, x_n +g) \end{align}
where $V$ is a real vector space of dimension $d$.
So for example if $d=2$ and we have $f(x,y)=x+y$ then the $\mathbb{Z}_3$ invariant version of $f$ might be
$$f_{inv}(x,y)= f(x,y)+f(x+1,y+1)+f(x+2,y+2) = 3x+3y$$
Is this correct?
