  In 2017 Davis, Nelson, Petersen, and Tenner pioneered the study of pinnacle
sets of permutations and asked whether there exists a class of operations,
which applied to a permutation in $\mathfrak{S}_n$, can produce any other
permutation with the same pinnacle set and no others. In this paper, we adapt a
group action defined by Foata and Strehl to provide a way to generate all
permutations with a given pinnacle set. From this we give a closed
non-recursive formula enumerating permutations with a given pinnacle set. Thus
answering a question posed by Davis, Nelson, Petersen, and Tenner.
