Suppose I have a morphism $f:X\to Y$ which is a GIT quotient of $X$ with respect to some reductive, linear group. Does the semistable $X^{ss}$ and stable locus $X^s\subset X$ determine completely the linearization (maybe up to taking a power of the linearization itself)?

Or, in better words, can two different linearizations $L$ and $L'$ of different GIT quotients of $X$ give the same semi-stable and stable locus?