A sigma, or skew, derivation is a natural generalisation of the notion of derivation depending on an algebra automorphism $\sigma$ which when equal to $id = \sigma$ reduces to the usual notion of a derivation. For a precise definition see here

https://planetmath.org/SigmaDerivation

Does there exist a notion of **skew differential graded algebra**
in the literature? If so where do these objects arise?

EDIT: To confirm I am asking if there exists a graded analogue of skew derivation algebra. So an $\mathbb{N}_0$-graded algebra $A = \bigoplus_{k \in \mathbb{N}_)0} A_k$, together with a degree $1$ map $d$ satisfying $d^2 = 0$, and a skew analogue of the graded Leibniz rule: $$ d(a \wedge b) = da \wedge \sigma(b) + (-1)^k \sigma(a)db, ~ a \in A_k $$