The Totient of a Square Is a Cube

Consider the number $50$ .
$50^{2} = 2500 = 2^{2} \times 5^{4}$ , so $ϕ (2500) = 2 \times 4 \times 5^{3} = 8 \times 5^{3} = 2^{3} \times 5^{3}$ . ¹
So $2500$ is a square and $ϕ (2500)$ is a cube.

Find the sum of all numbers $n$ , $1 < n < 10^{10}$ such that $ϕ (n^{2})$ is a cube.

¹ $ϕ$ denotes Euler's totient function.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.