Scaling the input vs scaling the impulse response for an LTI system Two different cases:

We pass $x(t)$ to an LTI system with impulse response $h(2t)$ and get the output $y(t)$.

We pass $x(2t)$ to an LTI system with impulse response $h(t)$ and get the output $z(t)$.


I'm looking for possible restrictions (e.g. on the input / impulse response) such that the $y(t)$ and $z(t)$ are equal, if possible.
So far I have this:
$$
y(t) = x(t) * h(2t) \implies Y(j\omega) = X(j\omega) \frac{1}{2} H(j\omega/2)
$$
$$
z(t) = x(2t) * h(t) \implies Z(j\omega) = \frac{1}{2} X(j\omega/2)H(j\omega) 
$$
If we take, for example, $X(j\omega)$ from the first equation and plug into the other one, we get
$$
Z(j\omega) = \frac{Y(j\omega/2)H(j\omega)}{H(j\omega/4)}
$$
I'm stuck at this point to comment on the relation between $y(t)$ and $z(t)$ and assert some restrictions to make them equal possibly.
Aside from

Input that is only a constant value or
a System that does nothing to the signal, i.e. $H = \text{const.}$,

these things are not the same.
Sketch of a proof:

Scaling input shifts energy to frequency components that hadn't energy before,
Scaling the system means changes the Transfer Function, but keeps the system LTI, which means it can't add energy to any components that had none before.