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Let $\mathcal M$ be a stack representing some moduli problem. Let $\mathcal X\to \mathcal M$ be the corresponding universal family.

What is the difference between the inertia stack $I\to \mathcal M$ and the universal automorphism group $\mathrm{Aut}_{\mathcal M}(\mathcal X)\to \mathcal M$?

I'm looking mainly at moduli problems of varieties, e.g., curves, polarized varieties, K3 surfaces, ppav's, etc. and I have the feeling the inertia stack and the universal automorphism group are canonically isomorphic. But I don't see why.

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