Let $(X,*)$ be a pointed topological space.

Let $F(X,k)/S_k$ be the $k$-th unordered configuration space.

Is there an inclusion $F(X,k)/S_k\to F(X,k+1)/S_{k+1}$ for each $k\geq 1$?

Note that $[x_1,\cdots,x_k]\mapsto [x_1,\cdots,x_k,*]$ is not well-defined.