Let $S$ be affine. Is there an example of a cokernel in $S$-group schemes, $$A \xrightarrow{\phi} B \xrightarrow{\beta} B/\phi(A)$$ which (a) require the group schemes to be thought of as sheaves in the fppf topology $(\text{Aff}_S)^{op}_{fppf} \to \text{Grp}$ for the sequence to be exact (and the sequence is not exact as sheaves over the etale or zariski site) and (b) doesn't have $\mu_d$ as the kernel.

The only examples satisfying (a) that I, and those around me, can come up with are the following:

 1. $\mu_{d} \to \mu_{n} \xrightarrow{z \mapsto z^d} \mu_{n/d}$; if $d \notin \mathcal{O}_S^*$ 
 2. $\mu_{d} \to \mathbb{G}_{m} \xrightarrow{z \mapsto z^d} \mathbb{G}_{m}$; if $d \notin \mathcal{O}_S^*$
 3. $\mu_{d}(k) \to SL_d(k) \to PGL_d(k)$; if char $k \mid d$

 




 
 

But these feel essentially the same. Are there any different examples?