Summary: Let \(a(x)\), \(b(x)\), \(p(x)\) be formal power series in the indeterminate \(x\) over \(\mathbb C\) such that \(\operatorname{ord} a(x)=0\), \(\operatorname{ord} p(x)=1\) and \(p(x)\) is embeddable into an iteration group \((\pi(s,x))_{s\in\mathbb C}\) in \(\mathbb C[[x]]\). By a covariant embedding of the linear functional equation
\[ \varphi(p(x))= a(x)\varphi(x)+b(x), \tag{L} \]
(for the unknown series \(\varphi(x)\in\mathbb C[[x]]\)) with respect to \((\pi(s,x))_{s\in\mathbb C}\) we understand families \((\alpha(s,x))_{s\in\mathbb C}\) and \((\beta(s,x))_{s\in\mathbb C}\) of formal power series which satisfy a system of cocycle equations and boundary conditions such that
\[ \varphi(\pi(s,x))= \alpha(s,x)\varphi(x)+ \beta(s,x), \quad s\in\mathbb C, \]
holds true for all solutions \(\varphi\) of (L). In this paper we present a complete solution of this problem and we demonstrate how earlier results concerning covariant embeddings with respect to analytic iteration groups can be derived from these more general results.