Determining size of symmetry groups with orbit-stabilizer theorem and Burnside's lemma.

Question

The classic orbit-stabilizer theorem and Burnside lemma problems tend to have the following structure:

1. Consider some object with a symmetry group, like a cube and its rotational symmetry group.
2. "Break" the symmetry of that object by imposing some additional structure on it, e.g. by coloring the faces of the previous cube.
3. Calculate the size of the symmetry group of the new object by applying Burnside's lemma and the orbit-stabilizer theorem.

Is there an algorithmic way to accomplish this, given that the problems tend to be generated algorithmically as above? For example, I can generate the problems:

How many ways are there to rotate a cube whose opposite faces are the same color that leave it invariant? (We break the symmetry of the cube by coloring faces)

How many ways are there to permute $X = (1,2,3,\ldots,n)$, $n$ an even number, such that only even numbers appear in even positions of the tuple? (We break the symmetry of the tuple by identifying even positions)

There are other problems that Burnside's lemma can be applied to, but I am specifically asking for advice on algorithmically solving problems of this type. Thanks!