Geodesic
Compatibility condition. The possibility of reduction to commutative situation
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 24-32

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There is an example of the nocompatibility in this paper for the odd case. Kernel and factor-group are $p$-groups. Besides there are some notes about lower examples for $p$-groups. 
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     author = {M. A. Bondarenko and B. B. Lur'e},
     title = {Compatibility condition. {The} possibility of reduction to commutative situation},
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     publisher = {mathdoc},
     volume = {443},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a2/}
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M. A. Bondarenko; B. B. Lur'e. Compatibility condition. The possibility of reduction to commutative situation. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 29, Tome 443 (2016), pp. 24-32. http://geodesic.mathdoc.fr/item/ZNSL_2016_443_a2/