  Stochastic symmetries and related invariance properties of finite dimensional
SDEs driven by general c\`adl\`ag semimartingales taking values in Lie groups
are defined and investigated. In order to enlarge the class of possible
symmetries of SDEs, the new concepts of gauge and time symmetries for
semimartingales on Lie groups are introduced. Markovian and non-Markovian
examples of gauge and time symmetric processes are provided. The considered set
of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by
L\'evy processes. Non trivial invariance results concerning a class of iterated
random maps are obtained as special cases.
