Some necessary conditions for compatibility of groups

Two groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> L₁ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> L₂ are compatible if there exists a finite group 𝐺 with isomorphic normal subgroups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> N₁ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> N₂ such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>≅</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> L₁ G/N₁ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>≅</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> L₂ G/N₂ . In this paper, we give new necessary conditions for two groups to be compatible.