In a recent paper with L. Q. Zamboni the authors introduced the class of $\vartheta$-episturmian words, where $\vartheta$ is an involutory antimorphism of the free monoid $A^{*}$. In this paper, we introduce and study $\vartheta$-characteristic morphisms, that is, morphisms which map standard episturmian words into standard $\vartheta$-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective.