  A loop is automorphic if all its inner mappings are automorphisms. We
construct a large family of automorphic loops as follows. Let $R$ be a
commutative ring, $V$ an $R$-module, $E=\mathrm{End}_R(V)$ the ring of
$R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for
every $a$, $b\in W$ and $1+a$ is invertible for every $a\in W$. Then
$Q_{R,V}(W)$ defined on $W\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an
automorphic loop.
  A special case occurs when $R=k<K=V$ is a field extension and $W$ is a
$k$-subspace of $K$ such that $k1\cap W = 0$, naturally embedded into
$\mathrm{End}_k(K)$ by $a\mapsto M_a$, $bM_a = ba$. In this case we denote the
automorphic loop $Q_{R,V}(W)$ by $Q_{k<K}(W)$.
  We call the parameters tame if $k$ is a prime field, $W$ generates $K$ as a
field over $k$, and $K$ is perfect when $\mathrm{char}(k)=2$. We describe the
automorphism groups of tame automorphic loops $Q_{k<K}(W)$, and we solve the
isomorphism problem for tame automorphic loops $Q_{k<K}(W)$. A special case
solves a problem about automorphic loops of order $p^3$ posed by Jedli\v{c}ka,
Kinyon and Vojt\v{e}chovsk\'y.
  We conclude the paper with a construction of an infinite $2$-generated
abelian-by-cyclic automorphic loop of prime exponent.
