How to intrude / extrude a cube (or other irregular shape (knife tool) onto a uniform shape such as a dodec or icos?

In this case a dodecahedron. Using the Regular Solid Tool, I can create a dodecahedron, but, at it's easiest, it then has to be able to rest flat on the grid, which, as it's designed isn't easy, without rotation. Same if I use "Regular Solid" inside Sverchok.

Next best thing is to create an icosahedron, level 1, bevel it, then merge distance to eliminate doubles.

Now I have a dodecahedron which I can rest the top or bottom flat on the grid by selecting the vertices of either the top or bottom face, moving cursor to selected, and in object mode, setting origin to cursor, moving cursor to origin, then moving selected object to cursor. (I believe I've got those steps right).

That would allow me to easily put a cube in either the top or bottom face using either a 4 faced cylinder which gives me one cube shape or a cube which gives me another.

That leaves ten faces on a dodecahedron.

Now, I've seen that with the knife tool, other "irregular" shapes can be used, but, the same problem remains. In this method, doing the top or bottom faces is easy, the same problem exists for the other ten.

There's Mesh tools and Inset which both allow for insetting a surface, but, no way to choose the number of sides for the inset. They only allow based on the existing faces for the inset which means for an Icosahedron, three sides or a Dodecahedron, five sides.

So, what would be the best, simplest way to get this done.

My example is a cube. A 4, 6, 7 or 8 sided circle would work just as well (knife tool(?) for example) as a uniform, if not "irregular" certainly other than the five side faces presented in a dodecahedron.

Example, using knife tool:

Here's my example to play with (4.2 MB):

https://pasteall.org/blend/efb28b3703124a388cdd7559bb6a3b48

This question, a good answer of which might apply here, was asked over two years ago:

How to snap an object onto a face of another object?

Face 2 Face Addon is well suited to this problem. (Not Free):

More here:

https://youtu.be/w5A_ob-Ovm4

as is Ran Tools - P-Cutter (also not Free):

More here:

https://www.blendermarket.com/products/rantools

Neither is prohibitively expensive.

There may be built in tools that work for this. I wasn't able to divine how to make best use of them.

In the absence of alternative explanations, there's these two tools. And, of the two of them, I found Face 2 Face much easier to use.

This is a very simple and elegant solution from a youtube tutorial, using Geometry nodes - 3.0: