If $c:\mathbb R\to c\mathbb R$ is a compactification of $\mathbb R$ then the cardinality $K=|c\mathbb R \setminus \mathbb R|$ of the remainder can be $\mathfrak c=2^{\aleph_0}$ or (at most) $|\beta \mathbb R\setminus \mathbb R|= 2^{\mathfrak c},$ and if $K<\mathfrak c$ then $K=1$ or $K=2$.
What other values of $K$ are consistently possible?
Let $S=\{t\in \mathrm{Card}: \mathfrak c\le t\le 2^{\mathfrak c}\}.$ Let $T$ be the set of uncountable cardinals of remainders of compactifications of $\mathbb R.$ Is it consistent that $T\ne S$? If so, is there any constraint on members of $S \setminus T$? Are there some forcing constructions that can address this?