  This work analyses types of group actions on families of $t$-dependent vector
fields of a particular class, the hereby called quasi-Lie families. We devise
methods to obtain the defined here quasi-Lie invariants, namely a kind of
functions constant along the orbits of the above-mentioned actions. Our
techniques lead to a deep geometrical understanding of quasi-Lie schemes and
quasi-Lie systems giving rise to several new results. Our achievements are
illustrated by studying Abel and Riccati equations. We retrieve the Liouville
invariant and study other new quasi-Lie invariants of Abel equations. Several
Abel equations with a superposition rule are described and we characterise Abel
equations via quasi-Lie schemes.

Classical Analysis and ODEs;Differential Geometry