Endomorphism kernel property for extraspecial and special groups

Abstract A group G has the endomorphism kernel property (EKP) if every congruence relation θ on G is the kernel of an endomorphism on G . In this note we fully characterize extraspecial groups which have EKP and show two series of special groups which have EKP, these series are specific semidirect products of types <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mrow> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:msub> <m:msup> <m:mrow/> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mfenced> <m:mo>⋊</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mtext> and </m:mtext> <m:msup> <m:mfenced open="(" close=")"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>×</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:mfenced> <m:mi>n</m:mi> </m:msup> <m:mo>⋊</m:mo> <m:msub> <m:mi>C</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> $\left(C_{p^2}{ }^n\right) \rtimes C_p \text { and }\left(C_p \times C_p\right)^n \rtimes C_p$ for all primes p and all n .