A class \(\mathcal X\) of finite groups is called a radical class or Fitting class, if the following two conditions are fulfilled: if \(G\in {\mathcal X}\) and \(N\triangleleft G\), then \(N\in{\mathcal X}\); if \(N_ 1,N_ 2 \in{\mathcal X}\) and \(N_ 1,N_ 2 \triangleleft G\), then \(N_ 1N_ 2 \in{\mathcal X}\). A subgroup \(V\) of a finite group \(G\) is called an \(\mathcal X\)-injector in \(G\), if \(K\cap V\) is the \(\mathcal X\)-maximal subgroup in \(K\) for every normal subgroup \(K\) in \(G\). L. A. Shemetkov has proposed to consider the question of existence of \(\mathcal X\)-injectors for radical classes \({\mathcal X} \subseteq {\mathcal G}\), where \(\mathcal G\) is the class of all solvable finite groups [Kourovka Notebook, Novosibirsk (1990; Zbl 0748.20001), Problem 11.117]. In the present article, a positive answer is obtained for \({\mathcal X} = {\mathcal G}_ \pi\), where \({\mathcal G}_ \pi\) is the class of all finite solvable \(\pi\)-groups.