Regular polygons constructed inside regular polygons

Let $P$ be a regular $n$-gon, and erect on each edge toward the inside a regular $k$-gon, with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.

Q1. Given $n$, for which $k$ is the center a regular polygon?

Here I believe the answer is: For odd $k \le \lfloor n/2 \rfloor$, the center is not a regular polygon; and otherwise it is. In the above example, $k=3,5$ lead to stars rather than regular polygons. But perhaps I am missing some cases?

Q2. When $n$ and $k$ lead to a central regular polygon, is it the case that the number of edges of the central polygon is always either $n$ or $2n$?

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Added. Here is $n=10$, $k=5$:
     

This answer is only a sketch; computations and formalities are left out. I assume $2<k<n$.

If $k$ is even then for each $k$-gon the edge nearest to the center of the $n$-gon is the one opposite the edge on the $n$-gon. This edge must therefore be on the boundary of the interior region. Each such nearest edge intersects the nearest edges of the neighbouring $k$-gons, hence the interior region is bounded by these nearest edges. By symmetry the interior region is then a regular $n$-gon.

If $k$ is odd and $2k<n$ then the centre of the regular $n$-gon is not contained in the regular $k$-gons, and the boundary of the interior region is not convex, and hence not a regular $m$-gon for any $m$.

If $k$ is odd and $2k=n$ then for each $k$-gon the vertex opposite its edge on the $n$-gon is the center of the $n$-gon. Hence there is no interior polygon.

If $k$ is odd and $2k>n$ then the center of the regular $n$-gon is contained in the regular $k$-gons. For each $k$-gon the edges nearest to the center of the $n$-gon are the ones adjacent to the vertex opposite the edge on the $n$-gon. Therefore these edges are on the boundary of the interior region. These nearest edges intersect one of the nearest edges of each of the neighbouring $k$-gons, hence the interior region is bounded solely by these pairs of adjacent edges. By symmetry the interior region is then a regular $2n$-gon

Summarizing, the interior region is:

• A regular $n$-gon if $k$ is even. In this case the apothem of the inner polygon is $a_i=|a_n-2a_k|$ where $a_n$ and $a_k$ are the apothems of the regular $n$-gon and $k$-gon, respectively.

• A regular $2n$-gon if $k$ is odd and $2k>n$. In this case the radius of the inner $2n$-gon is $r_i=r_k+a_k-a_n$, where $r_k$, $a_k$ and $a_n$ are the radius and apothem of the $k$-gons and the apothem of the $n$-gon.

• Not a regular polygon otherwise.