16. Let the function $f(x)=x^{2}-\left(k^{2}-5 a k+3\right) x+7(a, k \in \mathbf{R})$, given that for any $k \in[0,2]$, if $x_{1}, x_{2}$ satisfy $x_{1} \in[k, k+a], x_{2} \in[k+2 a, k+4 a]$, then $f\left(x_{1}\right) \geqslant f\left(x_{2}\right)$, find the maximum value of the positive real number $a$.
For each $k$, when $k^{2}-5 a k+3$
$=2 k+5 a$, $a$ reaches its maximum value.
That is, $5 a(k+1) \leqslant k^{2}-2 k+3$
$=(k+1)^{2}-4(k+1)+6$
$\Rightarrow 5 a \leqslant k+1+\frac{6}{k+1}-4$,
and $k+1+\frac{6}{k+1}-4 \geqslant 2 \sqrt{6}-4$,
with equality holding when $k=\sqrt{6}-1 \in[0,2]$,
thus $a \leqslant \frac{2 \sqrt{6}-4}{5}$. Therefore, the maximum value of the positive real number $a$ is $\frac{2 \sqrt{6}-4}{5}$.