Let $X$ be a random variable that follows a binomial distribution,
$$X \sim \text{Bin}(n,p)$$
and let $X$ be related to another random variable $Y$ so that $X=\xi(Y)$ and the inverse function exists, $Y=\xi ^{-1}(X)$.
In the case that $X$ is linearly related to $Y$, then will $Y$ also follow a binomial distribution? If so, how could the parameters of the new binomial distribution followed by $Y$ be found, $\text{Bin}(n_{_Y},p_{_Y})$? Would $Y$ also follow a binomial distribution in case the relation wasn't linear?
$Y$ need not have a Binomial distribution since its values need even be integers. Even if they are, the values need not be $0,1,2...,m$ for any integer $m$. We can only say $Y$ also take exactly $n+1$ values, namely $\xi^{-1}(i): 0\leq i\leq n$ and write down the corresponding probabilities.