  We investigate fixed-point properties of automorphisms of groups similar to
R. Thompson's group $F$. Revisiting work of Gon\c{c}alves-Kochloukova, we
deduce a cohomological criterion to detect infinite fixed-point sets in the
abelianization, implying the so-called property $R_\infty$. Using the BNS
$\Sigma$-invariant and drawing from works of Gon\c{c}alves-Sankaran-Strebel and
Zaremsky, we show that our tool applies to many $F$-like groups, including
Stein's $F_{2,3}$, Cleary's $F_\tau$, the Lodha-Moore groups, and the braided
version of $F$.
