The period-doubling sequence \((d_n)_{n\ge 0}\in\{0,1\}^{\mathbb{N}}\) is the fixed point of the morphism given by \(0\mapsto 01\) and \(1\mapsto 00\). It is quite a classical example of a 2-automatic sequence. Let \(D(X)=\sum_{n\ge 0} d_n X^n\) be the associated generating function. In this paper, the authors study its compositional inverse, i.e., a formal power series \(U(X)=\sum_{n\ge 0} u_n X^n\) such that \(U(D(X))=X=D(U(X))\). Following a method of M. Gawron and M. Ulas [Discrete Math. 339, No. 5, 1459–1470 (2016; Zbl 1338.11037)], they provide polynomial equations satisfied by \(U(X)\) and associated recurrence relations and show that the sequence \((u_n)_{n\ge 0}\) is \(2\)-automatic. Then the characteristic sequence of this formal inverse, i.e., the increasing sequence of indices such that \(u_n=1\), is shown to be \(k\)-regular for no \(k\). Surprisingly, Fibonacci numbers appear to be connected to these constructions.