R.T.~Curtis has defined the notion of a symmetric presentation: given a group K, permuting a set S, we consider a generating set X in bijection with S, with conjugation by K permuting X as K permutes S. A symmetric presentation is such a set up, together with relations given in terms of the elements of K and T.
It is not hard to see that, at least when K is transitive on S, this is equivalent to the set up for double coset enumeration, with one generator t, and gain group equal to the point stabiliser in K of some s_0 in S. The relations can be written in terms of K, t and conjugates of t by K, and so in terms of K and t.
GAP 3.4.4