This corresponds to a Steiner's Porism configuration with n = 4, however the trouble I'm having is that while it is easy to construct an n = 4 Steiner's Porism configuration (see second image below), I don't know what the circle of inversion would be that would invert it into the desired instantiation.
I was able to do some eyeballing (together with some observations such as that the outer 3 circles must be centered on the radical axes of pairs of the inner circles), using the excellent geometry software C.a.R. to snap to intersections, to construct an approximate diagram below, from which I was also able (approximately) to construct the circle of inversion $\omega$ (shown in red).
In black is shown the configuration as described, while in green are shown 4 of the 'regular 4-gon' figure resulting from inversion in $\omega$ (inner and outer concentric circles and two of the four congruent circles).
While the fact that this inversion works (again, approximately) shows that my eyeballing construction is reasonably close, I still would not know how to precisely construct $\omega$ directly from the regular 4-gon version of the n=4 configuration.
Also shown in light blue is a mid circle $\mu$ between two of the opposite circles in black of the described configuration.
This is problem 5.8.3 in Geometry Revisited (by Coxeter and Greitzer).