We investigate fixed-point properties of automorphisms of groups similar to Richard Thompson’s group $$F$$
F
. Revisiting work of Gonçalves and Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property $$R_\infty $$
R
∞
. Using the Bieri–Neumann–Strebel $$\varSigma $$
Σ
-invariant and drawing from works of Gonçalves–Sankaran–Strebel and Zaremsky, we show that our tool applies to many $$F$$
F
-like groups, including Stein’s group $$F_{2,3}$$
F
2
,
3
, cleary’s irrational-slope group $$F_\tau $$
F
τ
, the Lodha–Moore groups, and the braided version of $$F$$
F
.