A well known theorem by Scott says:

If $\kappa$ is a measurable cardinal and $\mu$ a normal measure on it and $\mu (\lbrace\lambda\in\kappa~|~2^{\lambda}=\lambda^{+}\rbrace)=1$ then $2^{\kappa}=\kappa^{+}$.

So:

If $\text{GCH}$ is false in a measurable cardinal then there is a smaller cardinal which $\text{GCH}$ is false in it too.

Equivalently:

If $\text{GCH}$ is false then its first failure point is not a measurable cardinal.

The role of existence of a normal measure on the cardinal $\kappa$ seems very essential in the proof of Scott's theorem. So perhaps one can prove the following statement:

For any large cardinal type $\text{A}$ smaller than measurables, it is consistent with $\text{ZFC}$ that $\text{GCH}$ be false and its first failure point be a large cardinal of type $\text{A}$.

Question is:

Question: Is the above statement true? For what type of large cardinals less than measurables is there a known result like above statement?