I am interested in regular graphs $G$ such that for each pair of 1-factors (=perfect matchings) $F$ and $F'$ there is an automorphism of $G$ that takes $F$ to $F'$. Let's call this property ***matching transitivity***.  

Those graphs should have a fairly big automorphism group. **I am wondering whether it is possible to characterize them.**  

For instance if we start with a matching-transitive graph (e.g. $K_{2n}$ or the complete bipartite graph $K_{n,n}$) and remove successively 1-factors from it, it is clear that this property is maintained at each stage. Note that if the graph has an odd cycle, we may arrive at a stage where the resulting graph has no more 1-factors, but then the notion is moot anyway, so we will exclude the graphs corresponding to those ultimate stages.

The question is inspired by the existence of graphs whose 2-factors are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type. For a discussion and examples of such graphs for various cycle types see [here][1] and [here][2]. All those graphs are cubic, and BTW I don't think that such a graph can be $k$-regular with $k>3$. (Is there an easy argument for that?)  
Some of those graphs have lots of symmetries, others a rather small automorphism group, like for instance, the graph of type $(5,11)$ given in [this answer][3], which has a unique triangle and automorphism group $S_3$, but not less than $12$ different 1-factors (so it cannot be matching-transitive). On the other hand, the [Heawood graph][4] ($|Aut(G)|=336$) is, and so is the [Coxeter Graph][5] (same automorphism group) as outlined [here][6].
  
So: 
> For given $n$ and $k$, can anything be said about the matching-transitivity of a $k$-regular graph on $n$ vertices in terms of the size (or structure) of its automorphism group?


  [1]: https://mathoverflow.net/questions/177625/cubic-graphs-whose-2-factors-all-have-the-same-cycle-type
  [2]: https://mathoverflow.net/questions/180299/more-3-connected-cubic-graphs-with-all-2-factors-of-same-cycle-type
  [3]: https://mathoverflow.net/a/177832/29783
  [4]: https://en.wikipedia.org/wiki/Heawood_graph#Combinatorial_properties
  [5]: http://mathworld.wolfram.com/CoxeterGraph.html
  [6]: https://www.famnit.upr.si/files/files/seminarji/coxeterpresentation-Anna%20Klymenko.pdf