  We show that several classes of groups G of PL-homeomorphisms of the real
line admit non-trivial homomorphisms from G to the additive group of reals that
are fixed by every automorphism of G. The classes of groups enjoying the stated
property include the generalisations of Thompson's group F studied by K. S.
Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups
investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that
every automorphism of a group in one of these classes has infinitely many
associated twisted conjugacy classes.
