Determining LTI system response to a scale change
I think it's instructive to look at this in the time domain. For LTI systems, the output $y(t)$ is given by the convolution of the input $x(t)$ with the impulse response $h(t)$:

$$y(t)=(x\star h)(t)=\int_{-\infty}^{\infty}x(t-\tau)h(\tau)d\tau\tag{1}$$

From $(1)$, the response to $x_a(t)=x(t-a)$, $a\in\mathbb{R}$, is given by $y(t-a)$:

$$\begin{align}(x_a\star h)(t)&amp;=\int_{-\infty}^{\infty}x_a(t-\tau)h(\tau)d\tau\\&amp;=\int_{-\infty}^{\infty}x(t-a-\tau)h(\tau)d\tau\\&amp;=y(t-a)\end{align}\tag{2}$$

which proves time-invariance.

However, the response to $x(at)$, $x\in\mathbb{R}$, is

$$\int_{-\infty}^{\infty}x(a(t-\tau))h(\tau)d\tau\neq y(at)\tag{3}$$

Consequently, in general we cannot deduce from the response $y(t)$ to an input $x(t)$ the response to a time-scaled input $x(at)$.