A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective if and only if $M$ is projective as an $A$ module via each map $f_i$.
For finite group schemes $G$ I will always denote $K[G]$ the commutative coordinate algebra and $KG$ its cocommutative dual, sometimes called the group algebra, sometimes the measure algebra. If $G$ is a finite (discrete) group, this means $K[G]$ has $|G|$ orthogonal idempotents, the coordinate algebra for a correspoding constant group scheme, while $KG$ is the usual group algebra with a basis of grouplike elements.
This question concerns only $B = KG$ for a finite group scheme $G$ and $A = K[t]/t^p$, where $K = \overline K$ has characteristic $p > 0$. For $E = (\mathbb{Z}/ p)^r$, by Dade's lemma, the set of embeddings for cyclic order $p$ subgroups is NOT big enough to detect projectivity for the group algebra $B = KE$. But there is another cocommutative Hopf algebra on $B =KE$, simultaneously the coordinate algebra for the Frobenius kernel $\mathbb{G}_{a(1)}^r$ and its dual, canonically a restricted Lie algebra. Projectivity not depending on Hopf algebra, this is to say the group $E$ is the wrong structure on $B$, as the other structure $B = K\mathbb{G}_{a(1)}^r$ produces a set of subgroups isomorphic to $\mathbb{G}_{a(1)}$, hence inclusion of group algebras isomorphic to a map $A \to B$; and it is this set $\mathbb{P}^{r-1}$ of 'generalized cyclic subgroups' which detects projectivity.
I think the theory of $\pi$-points [Friedlander and Pevtsova, $\Pi$-supports for modules for finite group schemes, 2007] lets us conclude that for $G$ a finite group scheme, $B = KG$, $G$ has projectivity detected by a set of subgroup schemes with group algebra $A$ if and only if $r + s = 1$ is the largest value of $r + s$ such that $G$ has subgroup scheme isomorphic to $\mathbb{G}_{a(r)}\times (\mathbb{Z} / p)^s$ (called quasi-elementary). It is easy to see if $kG$ is isomorphic as a Hopf algebra to the restricted enveloping algebra of any restricted lie algebra, that $G$ meets this condition.
Not every algebra occuring as a group algebra is also a restricted enveloping algebra like $(\mathbb{Z} / p)^r.$ For example take $p=2$ and $Q_8 = \{\pm1 , \pm i,\pm j, \pm k\}$, the quaternion group order 8 has no Lie algebra structure on $KQ_8$, and yet $B$ does have projectivity determined by the set of subgroups order $2$, the only one being $\langle -1 \rangle$.
Of discrete groups, there are no subgroups in the form $\mathbb{G}_{a(r)}$ for $r > 0$. The case of discrete groups overlaps with restricted Lie algebras at least for $G = \mathbb{Z}/(p^d)$, for which projectivity is detected by a single cyclic subgroup, and the inclusion of group algebras is identical to one in the form $A = K[\mathbb{G}_{a(1)}] \hookrightarrow K[\mathbb{G}_{a(d)}]$, the inclusion from a unique $\mathbb{G}_{a(1)}$ into a $d$ dimensional Lie algebra.
So we got Lie algebras and some special discrete groups like $Q_8$. What other types of finite group schemes $G$ (cocommutative Hopf algebras $KG$) meet the condition? I think it is reasonable we should only consider those group schemes with $KG$ having dimension a power of $p$ over $K$.
