= Solution
The action on the four diagonals identifies
$$
G\cong C_2\times S_4,
$$
where the $C_2$ factor is generated by central inversion. An element of order three must have trivial $C_2$ component and a three-cycle in $S_4$. Since all three-cycles in $S_4$ are conjugate,
$$
\boxed{\text{all elements of order three in }G\text{ are conjugate}.}
$$
Elements of order two are not all conjugate. For example, $-I$ is central and therefore has a singleton conjugacy class, while a reflection is a different element of order two. Thus
$$
\boxed{\text{elements of order two in }G\text{ need not be conjugate}.}
$$
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