Augmented Displayed Group

Let $G$ be a Group. An augmented displayed group over $G$ consists of:

• An H-Set $I:\mathrm{Set}$ of "-1 elements"
• An H-Set of objects $X:G\to I\to\mathrm{Set}$
• A multiplication operation $\cdot:X_{i,g_{1}}\to X_{i,g_{2}}\to X_{i,g_{1}g_{2}}$
• Identity elements $X_{i,0}$ for every $i$
• Inverses $(-)^{-1}:X_{i,g}\to X_{i,g^{-1}}$

such that the usual Displayed Group identities hold.

From the bundled perspective, this is a family of homomorphisms $X_{i}\to G$.