Square root of a matrix $A$ and matrices similar to $A$

Question

Prove or disprove:

$A \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries if and only if $XAX^{-1} \in \Bbb M(3,\mathbb{Z})$ has a square root with integer entries, for some invertible $X \in \Bbb M(3,\mathbb{R})$

Answer 1

If

$A = B^2, \; B \in M(3, \Bbb Z), \tag 1$

then for any invertible

$X \in M(3, \Bbb Z), \tag 2$

$XAX^{-1} = XB^2X^{-1} = (XBX^{-1})(XBX^{-1}) = (XBX^{-1})^2, \tag 3$

where

$XBX^{-1} \in M(3, \Bbb Z); \tag 4$

if

$XAX^{-1} = C^2, \; C \in M(3, \Bbb Z), \tag 5$

then

$A = X^{-1}C^2X = (X^{-1}CX)^2, \tag 6$

and evidently

$X^{-1}CX \in M(3, \mathbb Z). \tag 7$