If you are interested in constructing all 6-regular graphs on 20 vertices admitting a A5/icosahedral/dodecahedral vertex-transitive symmetry group, there is a very natural way to do this.
Let's take $G=A_5$ for example. Since $G$ will act faithfully and transitively on the 20 vertices, its action will be equivalent to the action of $G$ on the cosets of a subgroup of order 3. (This is a basic fact in permutation group theory).
Up to conjugacy, there is only one class of subgroups of order 3 in $G$, hence, up to equivalence, there is only one transitive action of $G$ on 20 points. Now, to reconstruct all the 6-regular graphs, all you have to do is find a set of sub-orbits with total size 6 and which is closed under "pairing". In fact, if you want $G$ to act arc-transitively, then you need to find a self-paired sub-orbit of length 6.
The same method can be applied to the icosahedral\dodecahedral groups.
If you are not familiar with this method, the keyword is "coset graph". (See http://www.sztaki.hu/~schneider/Teaching/4P4/chapter3.html for example).
Moreover, 20 is small enough that, with a little more work, it is possible to find ALL the symmetric 6-valent graphs on 20 vertices.
In fact, all the vertex-transitive graphs up to 32 vertices of any valency are known!
(See http://symomega.wordpress.com/2012/02/27/there-are-677402-vertex-transitive-graphs-on-32-vertices/)
You could then simply work through this list, find all 6-valent vertex-transitive graphs on 20 vertices, filter out those which are not edge-transitive (or arc-transitive, you were not very clear) and compute their chromatic numbers!
EDIT: I went ahead and checked Gordon's data. Out of the 80 connected 6-valent vertex-transitive graphs on 20 vertices, only 5 are also edge-transitive. (They are also arc-transitive).
One of them is a Circulant, three are Cayley graphs on F20 (the Frobenius group of order 20) and the last one is not a Cayley graph. They all have girth 3 or 4. I was only able to compute the chromatic number in one case.
Here they are in graph6 format, with a summary of the data. I can post the adjacency matrices if you prefer that.
SsaCBLYNAWEOP@Q@@_`CRCagoJ?Bf?B_w
(10x2):D6
Circulant
Girth 4
SsaCBLYNBOI_[?I_Ao?[??Mk@VOBZ??^_
S5x2
Non-Cayley
Girth 4
SsaCB|_WB?K?EKEKB@_oW@kKEooK]?K]?
2^10:S5,
Cayley on F20
Girth 4
S{aSQGGhI?oE@OpGc_eIAgROgXQ_B{?
S5x2
Cayley on F20
Girth 3
S~aKYPDHGqCQCbOWCAg_VAQ?CoGCKW?B{
S5x2
Cayley on F20
Girth 3
Chromatic Number 4.
The second line is a rough description of the automorphism group. The first one has a solvable group, so will not have the groups you are interested in. All the others contain copies of A5.