Diffeotopy group, Mapping Class group, Isometry group

Question

There are several closely related concepts on the symmetries or symmetry groups of the space.

My apology, but some vague imprecise definitions may be as:

1. Mapping class group (MCG) is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to the symmetries of the space.

2. Diffeotopy group: The diffeotopy group D(M) of a smooth manifold M is the quotient of the diffeomorphism group Diff(M) by its normal subgroup Diff$_0$(M) of diffeomorphisms isotopic to the identity. Alternatively, one may think of the diffeotopy group as the group $π_0$(Diff(M)) of path components of Diff(M), since any continuous path in Diff(M) can be approximated by a smooth one, i.e. an isotopy.

3. Isometry group: the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.

These concepts are important and relevant for exotic $\mathbb{R}^4$, for example, we can ask: "Does every isometry group of an exotic $\mathbb{R}^4$ inject into its diffeotopy group?"

So my question here is aimed for a relation and interpretations between the definitions of Diffeotopy group, Mapping Class group, Isometry group?

For example, it seems that $$ \text{MCG($S^2 \times S^1)= \mathbb Z_2 \oplus \mathbb Z_2$} $$ $$ \text{extended-MCG($S^2 \times S^1)=\mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2$} $$ similarly,

$$ \text{ diffeotopy group of $(S^2 \times S^1)=\mathbb Z_2 \oplus \mathbb Z_2 \oplus \mathbb Z_2$} $$

Are there general relations between the above Diffeotopy group, Mapping Class group, Isometry group?