A Fitting class construction is presented which has affinities with that of \textit{R. S. Dark} [Math. Z. 127, 145-156 (1972; Zbl 0226.20013)]. All groups considered are finite and soluble. Let \(p\), \(q\), \(r\) be primes. The author defines \({\mathcal M}\) as the class of groups of the form \(ABU\), where \(A\), \(B\) and \(U\) are nontrivial subgroups, with the following properties: i) \(U=O_ r(ABU)\), ii) \(U/\Phi(U)\), the Frattini factor group of \(U\), is minimal normal in \(ABU/\Phi(U)\), iii) \(C_{ABU}(U/\Phi(U))=U\), iv) \(B\) is a (nontrivial) Sylow \(q\)-subgroup of \(ABU\), and \(q\neq r\), v) \(B\trianglelefteq AB\), vi) \(B/\Phi(B)\), the Frattini factor group of \(B\), is minimal normal in \(AB/\Phi(B)\), vii) \(C_{AB}(B/\Phi(B))=B\), viii) \(A\cong C_ p\), with \(p\neq q\), ix) either \(U/\Phi(U)\) is minimal normal in \(BU/\Phi(U)\) or no two minimal normal subgroups of \(BU/\Phi(U)\) are operator-isomorphic (with regard to \(B\)). The class \({\mathcal F}\) of groups consists of all those \(ABU\in {\mathcal M}\) such that for any group \(H=H_ 1H_ 2\) where \(H_ 1\) and \(H_ 2\) are normal in \(H\) and \(W_ 1=O^ p(O^{p'}(H_ 1))=\langle B_ 1U_ 1,B_ 2U_ 2,\dots,B_ kU_ k,R_ 1\rangle\), \(W_ 2=O^ p(O^{p'}(H_ 2))=\langle D_ 1V_ 1,D_ 2V_ 2,\dots,D_ lV_ l,R_ 2\rangle\), for suitable natural numbers \(k\) and \(l\), the subgroups \(B_ 1U_ 1,B_ 2U_ 2,\dots,B_ kU_ k\) and \(D_ 1V_ 1,D_ 2V_ 2,\dots,D_ lV_ l\) are all normal in \(O^{p'}(H)\) as long as the following conditions are met: letting \(i\) range from 1 to \(k\) and \(j\) from 1 to \(l\), 1) \(B_ iU_ i\cong D_ jV_ j\cong BU\), \(B_ i\cong D_ j\cong B\) and \(U_ i\cong V_ j\cong U\), 2) \(B_ iU_ i\trianglelefteq O^{p'}(H_ 1)\) and \(D_ jV_ j\trianglelefteq O^{p'}(H_ 2)\), 3) \(U_ i/F(U_ i)\) is minimal normal in \(O^{p'}(H_ 1/F(U_ i))\) and \(V_ j/F(V_ j)\) is minimal normal in \(O^{p'}(H_ 2)/F(V_ j)\), 4) \(R_ 1=O_ r(W_ 1)\) is the normal Sylow \(r\)-subgroup of \(W_ 1\) and \(R_ 2=O_ r(W_ 2)\) is the normal Sylow \(r\)-subgroup of \(W_ 2\), 5) \(O_{r'}(W_ 1)=1=O_{r'}(W_ 2)\), 6) \(W_ 1=B_ iC_{W_ 1}(U_ i/F(U_ i))\) and \(W_ 2=D_ jC_{W_ 2}(V_ j/F)V_ j))\). Finally, \({\mathcal FK}\) is defined as the class of groups \(ABUR\) with i) \(R\trianglelefteq ABUR\) and \(R\cap ABU=1\), ii) \(ABU\in{\mathcal F}\) (with \(A\), \(B\), and \(U\) as above), iii) \(R=(ABUR)_{\mathcal T}\), the \(\mathcal T\)-radical of \(ABUR\), where \(\mathcal T\) is a Fitting class which does not contain \(q\)-groups (\(q\) as above), iv) \(O^{q'}(BUR)=BUR\). The main result is: Let \(G=ABUR\) be a (fixed) \(\mathcal FK\)-group. Let \({\mathcal X}_ G\) be the class of groups, \(H\), which have the properties that i) for a suitable natural number \(k\), \(O^ p(O^{p'}(H))=\langle Y_ 1,\dots,Y_ k,D\rangle\), where \(D\) is defined by \(N=O^ p(O^{p'}(H))_{\mathcal T}\leq D\) (\(\mathcal T\) as above) and \(D/N=O_ r(O^ p(O^{p'}(H))/N)\), and letting \(i=1,\dots,k\), ii) \(Y_ i\trianglelefteq O^{p'}(H)\), iii) \(Y_ i\cong BUR\), iv) for \(R_ i=(Y_ i)_{\mathcal T}\) and \(U_ i/R_ i=O_ r(Y_ i/R_ i)\) and \(\Phi_ i/R_ i=\Phi(U_ i/R_ i)\), we have \(O^{p'}(H)/C_ i=(Y_ iC_ i/C_ i)P_ i\), where \(C_ i=C_{O^{p'}(H)}(U_ i/F_ i)\) and \(P_ i\) is a Sylow \(p\)-subgroup of \(O^{p'}(H)/C_ i\), v) for \(B^*_ i=Y_ iC_ i/C_ i\) we have \(B_ i^*P_ i/C_{P_ i}(B_ i^*)\cong AB\), and vi) \(O_ q(O^{p'}(H)/N)=1\); then the class \({\mathcal X}_ G\) is a Fitting class containing \(G\).