Revised answer (see previous version to make sense of comments):
The Petersen graph is a small wonderful and rather exceptional creature. It is not a paradigm.
Vary your construction:
For $G = C_m$, $H = K_2$ and
- $\pi_1(i) = \text{id}$ for $i=1,\dots,m$
- $\pi_2(1) = \text{id}$
- $\pi_2(2) = (12)(3\cdots m)$ link text I don't think that the result is vertex transitive so your construction need not (and probably seldom) gives a vertex transitive result.
There are many vertex transitive graphs with a prime number of vertices, none of them is a product. So there are graphs which do not arise in this manner.
What can you get? Can you get $K_{2m}$ minus a matching for $m=3$, or even any $m \ge 3$?