Permute association key and generate a sparse array

Maybe with SymmetrizedArray?

size = 4;
rank = 4;
Av = <|{1, 4, 2, 3} -> a, {2, 4, 2, 4} -> b|>;
symmetries = {
   {{4, 2, 3, 1}, 1},
   {{4, 3, 2, 1}, 1},
   {{1, 3, 2, 4}, 1},
   {{2, 1, 4, 3}, 1},
   {{3, 1, 4, 2}, 1},
   {{3, 4, 1, 2}, 1},
   {{2, 4, 1, 3}, 1}
   };
A = SparseArray[
  SymmetrizedArray[Normal[Av], ConstantArray[n, rank], symmetries]
  ]

In the list symmetries, each entry is a pair {p,s} of a permutation p that can be applied and a sign s (1, -1) that tells us whether the array is meant to be symmetric (s = 1 means symmetric, s = -1 means antisymmetric).

In fact, it suffices to provide only a set of generators of the symmetry group. For example,

symmetries = {
   {{4, 2, 3, 1}, 1},
   {{1, 3, 2, 4}, 1},
   {{2, 4, 1, 3}, 1}
   };

would lead to the same result

pg = PermutationGroup[{{1, 2, 3, 4}, {4, 2, 3, 1}, {4, 3, 2, 1}, {1, 
     3, 2, 4}, {2, 1, 4, 3}, {3, 1, 4, 2}, {3, 4, 1, 2}, {2, 4, 1, 3}}];

ClearAll[sAbuild]
sAbuild = SparseArray[KeyValueMap[Alternatives @@ 
         GroupOrbits[pg, {#}, Permute][[1]] -> #2 &]@#, #2] &;

dims = {4, 4, 4, 4};

sAbuild[Av, dims]

Based on Henrik's answer, we can use a smaller set of generators to get the same result:

pg = PermutationGroup @ {{4, 2, 3, 1}, {1, 3, 2, 4}, {2, 4, 1, 3}};