Let $A/k$ be an abelian variety over an algebraically closed field and $\ell \neq \mathrm{char}\,k$.

In http://jmilne.org/math/articles/1986b.pdf, Theorem 15.1(b) it is proved that $$H^r_{et}(A, R) = \bigwedge^rH^1_{et}(A,R)\quad\text{for $R = \mathbf{Z}_\ell,\mathbf{Q}_\ell,\mathbf{F}_\ell$.}$$ The proof uses $$H^1_{et}(A,\mathbf{Z}_\ell) = \mathrm{Hom}_{cont}(\pi_1^{et}(A,0),\mathbf{Z}_\ell).$$

For $R = \mathbf{Z}/p\mathbf{Z}$ one has $$H^r_{et}(A,\mathbf{Z}/p\mathbf{Z}) = H^r_{fppf}(A,\mathbf{Z}/p),$$ since $\mathbf{Z}/p\mathbf{Z}$ is a smooth quasi-projective commutative group scheme. Using induction, the five lemma and short exact sequences $$0 \to \mathbf{Z}/p \to \mathbf{Z}/p^{n+1} \to \mathbf{Z}/p^n \to 0,$$ one gets $$H^r_{et}(A,\mathbf{Z}/p^n) = H^r_{fppf}(A,\mathbf{Z}/p^n)\quad\text{ for all $n \geq 0$,}$$ and hence $$H^r_{et}(A,\mathbf{Z}_p) = H^r_{fppf}(A,\mathbf{Z}_p).$$

One has $$\pi_1^{et}(A,0) = \prod_{\ell} T_\ell(A)$$ with $T_p(A) = \varprojlim_nA[p^n]^r$ with $A[p^n] = A[p^n]^0 \times A[p^n]^r$, with $A[p^n]^0$ local and $A[p^n]^r$ reduced (Mumford, *Abelian Varieties*, Chapter IV.18) for $p = \mathrm{char}\,k$.

By Lei Fu, *Étale cohomology theory*, Proposition 5.7.20, one has $$H^1_{et}(X,G) = \mathrm{Hom}_{cont}(\pi_1^{et}(X),G)$$ for a finite group $G$ and $X$ connected Noetherian.

Is there a similar isomorphism for crystalline cohomology or flat cohomology with $R = \mathbb{Z}_p,\mathbb{Q}_p,\mathbb{F}_p$, $p = \mathrm{char}\,k$?

**Edit (26.04.2018):** The answer below settles the question for $W(k)$ and $\mathrm{Quot}(W(k))$ coefficients and crystalline cohomology.

What is the analogue of $H^q_\mathrm{et}(\bar{A},\mu_{\ell^n}) = \mathrm{Hom}(\bigwedge^qA[\ell^n],\mu_{\ell^n})$ in crystalline cohomology?