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Quotient of a reductive group by a non-smooth central finite subgroup

I need a construction in linear algebraic groups which uses taking quotient by a central finite group subscheme. My question is, whether it goes through in bad'' characteristics, when this group subscheme is not smooth. First I write this construction in a special case, and then in the general case.

Let $G$ be a connected semisimple $k$-group over a field $k$ of characteristic $p>0$. We may assume that $k$ is algebraically closed. Assume that the corresponding adjoint group $G^{ad}$ is $PGL_n$.

In general, my group $G$ is not special (recall that a $k$-group $G$ is called special if $H^1(K,G)=1$ for any field extension $K/k$). I want to construct a special $k$-group $H$ related to $G$. For this end I consider the universal covering $G^{sc}$ of $G$, then $G^{sc}=SL_n$. Let $Z$ denote the center of $G^{sc}$, then $Z=\mu_n$.

We have a canonical epimorphism $\varphi\colon SL_n \to G$. We denote by $C$ the kernel of $\varphi$. Then $C$ is a group subscheme of $Z$, defined over $k$.

Since $Z=\mu_n$, there is a canonical embedding $Z\hookrightarrow \mathbb{G}_m$ into the multiplicative group $\mathbb{G}_m$. Thus we obtain an embedding $C\hookrightarrow \mathbb{G}_m$. Consider the diagonal embedding $$C\hookrightarrow SL_n\times \mathbb{G}_m.$$ I would like to define $H:=(SL_n\times \mathbb{G}_m)/C$. Is such a quotient defined, when char($k$) divides $n$ and $C$ is not smooth?

Note that $SL_n$ embeds into $H$, and we have a short exact sequence $$1\to SL_n \to H \to \mathbb{G}_m \to 1$$ In this exact sequence both $SL_n$ and $\mathbb{G}_m$ are special, and from the Galois cohomology exact sequence we see that $H$ is special as well.

In the general case I assume that $G^{ad}$ is a product of groups $PGL_{n_i}$, $i=1,\dots s$. Then $G^{sc}$ is the product of $SL_{n_i}$. Let $C$ denote the kernel of the canonical epimorphism $\varphi\colon G^{sc}\to G$, then $C$ is contained in the center $Z$ of $G^{sc}$. We have $Z=\prod_{i=1}^s \mu_{n_i}$. Again we embed diagonally $$C\hookrightarrow (\prod_{i=1}^s SL_{n_i}) \times (\mathbb{G}_m)^s$$ and denote by $H$ the quotient. Again $H$ is special (if it is defined), and again my question is, whether this construction makes sense when char($k$) divides $n_i$ for some $i$.

Any help is welcome!