How to find the impulse response of filter that outputs the autocorrelation of the input? I am trying to solve question 9.61 in the attached image.  



I guess, by evaluating the autocorrelation for a specific instance,  say $ t = 0 $ , that the impulse response, $h(t)$, should be : $$ h (t) =  x(-t) $$.

But can I get this from equating the convolution integral to the given output integral, i.e.

$$y(t) = \int_{-\infty}^{\infty} x(\tau)x(t + \tau) d\tau = \int_{-\infty}^{\infty} x(\tau)h(t - \tau) d\tau $$ .

Is it then right to equate as below?

$$ x(t + \tau) = h (t - \tau) $$

Thank you!
HINT:

Note that

$$\int_{-\infty}^{\infty}x(\tau)x(t+\tau)d\tau=\int_{-\infty}^{\infty}x(\tau-t)x(\tau)d\tau$$