Find the Impulse response given input and output signals
Here is one way to find the answer. Uses two properties of linear time invariant systems.

if \$ x(t) \mapsto y(t)\$, then the same system does : \$\frac{d x(t)}{dt} \mapsto \frac{d y(t)}{dt}\$.
If the input is sum of delayed copies of a particular signal (say impulse), then output is also made of equally delayed copies of some signal. This is the time invariance property. If \$u(t) \mapsto v(t)\$, then
\$u(t+\tau_1) + u(t+ \tau_2) + \dots \mapsto v(t+\tau_1) + v(t+\tau_2) + \dots\$.

With these two properties in mind, see the below diagram. The top left input gives top middle output. Using property #1, we know that, for the same system, bottom left input gives bottom middle signal as output.
Now using property #2, we need to split the bottom middle picture into two copies of same signal delayed by 2T since input is made from two impulses spaced by 2T. Here one of the copy is negated since the input also has a negative impulse.

But how can we split the bottom middle signal into two pieces correctly ? Let's use the properties again.
The input impulse \$-\delta(t-T)\$ cannot generate the output impulse \$\delta(t-2T)\$ (labelled A). If it did, then the impulse \$\delta(t+T)\$ should have generated an impulse at -4T, \$-\delta(t+4T)\$ by property #2. So the impulse labelled A &quot;belongs&quot; to \$\delta(t+T)\$. This means that one of the two impulses at 0, \$-\delta(t+0)\$ (labelled C or D) &quot;belongs&quot; to \$-\delta(t-T)\$. Iteratively apply this same rule from left to right until all the impulses in the output are accounted for.
Once this is done, it will be possible to synthesize an output for the input \$\delta(t+0)\$.