Let $X$ be a projective algebraic variety over a (perfect) field $k$.

Let $Aut(X):k \text{-}Alg \to Grp$ be the functor of points defined by

$$Aut(X) : A \mapsto Aut_{Spec (A)}(X \times_{k} Spec (A))$$

In this situation $Aut(X)$ is representable by an open subscheme of the Hom scheme $\mathsf{Hom}(X,X)$. It is therefore a group scheme locally of finite type over $k$.

Question 1: Can $Aut(X)$ be non-reduced? ($k$ has positive characteristic here of course) if so, what's an example of this (preferably one where $X$ is itself reduced)?

Let $Aut_0(X)$ be the connected component of the identity. As far as I understand the functor of points of the quotient $Aut(X)/Aut_0(X)$ is also representable by a scheme at this level of generality, lets denote it by $MCG(X)$.

Question 2: Is $MCG(X)(k)$ finitely presented as an abstract group?

Finally, are there any non-trivial examples for which $MCG(X)$ has been calculated?