Here's a snippet of GAP code that can search for the groups you request (it uses the SONATA package):
RequirePackage("sonata"); for p in Filtered([2..5],IsPrimeInt) do for r in Filtered([2..5],i->(not IsPrimePowerInt(i)) or i mod p<>0) do n:=p^2*r; for G in AllSmallGroups(n) do S:=Filtered(Subgroups(G),H->Size(H)=p); if(Size(S)=p+1 and Size(SylowSubgroup(G,p))=p^2) then Print("Group: ",StructureDescription(G)," of order ",Size(G)," has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=",p,"\n"); fi; od; od; od;
This code implicitly uses the property that (a) all groups of prime order a cyclic, and (b) Lagrange's Theorem (so $p^2$ must divide the order of the group).
Here's the output:
Group: A4 of order 12 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=2 Group: C6 x C2 of order 12 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=2 Group: C10 x C2 of order 20 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=2 Group: C3 x S3 of order 18 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: (C3 x C3) : C2 of order 18 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C6 x C3 of order 18 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C3 x (C3 : C4) of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: (C3 x C3) : C4 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C12 x C3 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: (C3 x C3) : C4 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: S3 x S3 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C6 x S3 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C2 x ((C3 x C3) : C2) of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C6 x C6 of order 36 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 Group: C15 x C3 of order 45 has p+1 subgroups of order p and Sylow p-subgroup of order p^2, where p=3 [[snipped]]
(Here ":" denotes a semidirect product.)