  In this note, we compute the {\Sigma}^1(G) invariant when 1 {\to} H {\to} G
{\to} K {\to} 1 is a short exact sequence of finitely generated groups with K
finite. As an application, we construct a group F semidirect Z_2 where F is the
R. Thompson's group F and show that F semidirect Z_2 has the R-infinity
property while F is not characteristic. Furthermore, we construct a finite
extension G with finitely generated commutator subgroup G' but has a finite
index normal subgroup H with infinitely generated H'.

Group Theory