  We explore algorithmic aspects of a simply transitive commutative group
action coming from the class field theory of imaginary hyperelliptic function
fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over
$\mathbb F_q$ acts on a subset of isomorphism classes of Drinfeld modules. We
describe an algorithm to compute the group action efficiently. This is a
function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We
report on an explicit computation done with our proof-of-concept C++/NTL
implementation; it took a fraction of a second on a standard computer. We prove
that the problem of inverting the group action reduces to the problem of
finding isogenies of fixed $\tau$-degree between Drinfeld $\mathbb
F_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm by
Wesolowski. We give asymptotic complexity bounds for all algorithms presented
in this paper.
