Abstract Let $${{\mathcal {S}}}{{\mathcal {S}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> and $${{\mathcal {S}}}{{\mathcal {C}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let $$P\Phi _+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:math> and $$P\Phi _+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>P</mml:mi> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:math> be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators $$\Phi {\mathcal {S}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> and $$\Phi {\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Φ</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> that satisfy $${{\mathcal {S}}}{{\mathcal {S}}}\subset \Phi {\mathcal {S}}\subset P\Phi _+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>S</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Φ</mml:mi> <mml:mi>S</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>P</mml:mi> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:mrow> </mml:math> and $${{\mathcal {S}}}{{\mathcal {C}}}\subset \Phi {\mathcal {C}}\subset P\Phi _-.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>C</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Φ</mml:mi> <mml:mi>C</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>P</mml:mi> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mo>-</mml:mo> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> We give some conditions under which these inclusions become equalities, from which we derive some positive solutions to the perturbation classes problem for semi-Fredholm operators.