The answer to all three questions is yes and certainly is classical.

One simple example is the following:

Let $C_2$ act on a faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivial element of $C_2$ swaps 1,2 and also swaps 3,4. In the second the non-trivial element swaps 1,2 and fixes 3,4.

Each action defines a representation of $C_2$ on $\mathbb{Z}^4$ via permuation matrices. In one case the trace of the non-trivial permutation matrix is $4$ in the other $2$ so the images cannot be conjugate in $GL_4(\mathbb{Z})$ or $GL_4(\mathbb{R})$ however they both generate a subgroup $C_2$ in the former.

It is fairly clear this idea generalises to any isomorphism class of finite groups.

The answer to all three questions is yes and certainly is classical.

One simple example is the following:

Let $C_2$ act faithfully on the set $\{1,2,3,4\}$ in two ways. In the first the non-trivial element of $C_2$ swaps 1,2 and also swaps 3,4. In the second the non-trivial element swaps 1,2 and fixes 3,4.

Each action defines a representation of $C_2$ on $\mathbb{Z}^4$ via permuation matrices. In one case the trace of the non-trivial permutation matrix is $0$ in the other $2$ so the images cannot be conjugate in $GL_4(\mathbb{Z})$ or $GL_4(\mathbb{R})$ however they both generate a subgroup $C_2$ in the former.

It is fairly clear this idea generalises to any isomorphism class of finite groups.