I would like to know how to link a scaled beta distribution to its non scaled beta distribution.

For some context: I have a model where I want to estimate the proportion ($\pi$) of people belonging to a given group (out of 2 groups) (can see this as the probability that a given event happen).
To estimate the true proportion $\pi$ I would simply do bayesian updating based on a beta prior that $\pi \sim Beta(a, b)$, with binomial (bernouilli) likelihood $f(x|\pi)$ for the data.

However, I do not observe the belonging to one or the other group directly in my data. What I observe from the data is the belonging to two different groups, where the proportion of people belonging to my group of interest is $c \ \pi$ (where $c$ is a known constant $\in [0, 1]$). Hence my data follow a bernouilli likelihood: $c \pi^x \ (1-c\pi)^{1-x}$.

You probably wonder, why don't I simply put the beta prior on $c \pi$ instead and then just divide the expectation to find an estimation of $\pi$? I cannot because $c$ can vary every period (and because I want $\pi$ to belong to $[0,1]$). Given that $c$ can vary each period, I want to update my prior on $\pi$ and not on $c \pi$, and do the link between the two no matter what $c$ is (and anyway, my parameter of interest is actually $\pi$).

I am quite new to bayesian updating of binomial proportion and Beta distribution in general, and I was not able to find an answer to this. I know that if $\pi \sim Beta(a,b)$, then $c \pi$ is not a beta distribution anymore but a "scaled beta distribution" (is there any other name? I do not care about the whole generalized beta distribution, I just want the additional scale parameter!).
To solve my problem I just need to know how to link two scaled beta distribution (with the same a and b) (actually, how to link a scaled beta distribution to its version with scale = 1).

From what I tried so far, I am stuck because I end up with: $something \times (c \pi)^x (1-c\pi)^{1-x} \pi^{a-1} (1-\pi)^{b-1}$ which does not allow me to do simple updating and say for example that it is obviously another beta distribution (as we do in the bayesian updating classic case).

I do not see an easy factorization here, but maybe I just miss a simple maths trick? or maybe my formula is even wrong?
My main question is, is there an easy way to update "scaled beta distribution" (or beta of the first kind) or do I have to do it numerically?


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