  In this article, we study the cyclicity problem of elliptic curves
$E/\Bbb{Q}$ modulo primes in a given arithmetic progression. We extend the
recent work of Akbal and G\"ulo\u{g}lu by proving an unconditional asymptotic
for such a cyclicity problem over arithmetic progressions for CM elliptic
curves $E$, which also presents a generalisation of the previous works of
Akbary, Cojocaru, M.R. Murty, V.K. Murty, and Serre. In addition, we refine the
conditional estimates of Akbal and G\"ulo\u{g}lu, which gives log-power savings
(for small moduli) and consequently improves the work of Cojocaru and M.R.
Murty. Moreover, we study the average exponent of $E$ modulo primes in a given
arithmetic progression and obtain several conditional and unconditional
estimates, extending the previous works of Freiberg, Kim, Kurlberg, and Wu.
