Extreme Value Theory Quantile Estimation

Question

I have been reading Coles' and De Haan's books on Extreme Value Theory, and the suggestion is made as a starting point that one can start with a sample, break it into blocks, find the maximum of each block, fit these to a Generalized Extreme Value Distribution using one of several options for parameter estimation (I just used MLE as a starting point). An application seems to be estimating quantiles for very unlikely extreme events. That is: it is suggested in Coles' book that the quantile for the $p=1/100$ threshold for the original distribution can be estimated by the $p=1/100$ threshold for the extreme value distribution. That is to say, if $\hat{G}$ is the MLE for the GEVD associated to $M_n=\max_{i\leq n}\{X_i\}$ and $F_{X_i}$ the CDF for X_i, then $$F_{X_i}^{-1}(1-p)\approx \hat{G}^{-1}(1-p).$$ It's not obvious to me at all based on what I'm seeing in the book if this is the case or why it would be the case. In computing this for 1000 samples of standard exponentially distributed R.V. with block size 25, I recovered $\hat{G}^{-1}(1-\frac{1}{100})\approx7,$ whereas the actual answer is $F^{-1}(.99)=-\log(.01)\approx 4.6...$ I feel like I must be missing something fundamental.