Here are some aspherical examples.
Suppose that $M$ is a closed hyperbolic (real or complex) manifold with 1-st Betti number $>1$ (such exist in abundance); I will assume that the real dimension of $M$ is $>2$. Then the group of outer automorphisms of $\pi_1(M)$ is finite (by the Mostow Rigidity Theorem), while the group of automorphisms of the 1st homology group is infinite. The outer automorphism group $Out(\pi_1(M))$ is the same as the group of self-homotopy-equivalences of $M$ due to asphericity of $M$.
To make things more interesting, take a 4-dimensional closed real-hyperbolic manifold with $b_2(M)>0$ (examples, again, abound). Since $M$ has zero signature, the group of automorphisms of the intersection form on $H_2(M)$ is infinite, while $Out(\pi_1(M))$ is again finite.
Edit. Here are some details:
Definition. A group $G$ is called Hopfian if each epimorphisms $G\to G$ is an isomorphism.
Malcev proved in 1940 that all finitely-generated matrix groups are Hopfian. (He noticed that each residually finite group is Hopfian and then proved that all finitely generated matrix groups are residually finite.) I will use this in the context of fundamental groups of complete hyperbolic and complex-hyperbolic manifolds. These groups are subgroups of $PO(n,1)$ and $PU(n,1)$ respectively (where $n$ is the real and complex dimension respectively). For instance, if $M$ is a hyperbolic n-manifold, then $M=H^n/G, G<PO(n,1)$, $H^n$ is the hyperbolic n-space.
Lemma. Suppose that $M$ is a closed oriented manifold with Hopfian fundamental group. Let $f: M\to M$ be a degree $\pm 1$ map. Then $f$ induces an automorphism of $\pi_1(M)$.
Proof. Let $G=\pi_1(M)$, $H=f_*(G)<G$. If $H\ne G$, then the map $f$ lifts to a map $f': M\to M'$, where $p: M'\to M$ is the covering corresponding to the subgroup $H$. Since $deg(f)=deg(f')deg(p)$ (with both sides equal to zero if $p$ has infinite degree), we obtain a contradiction. Hence, $H=G$. Since $G$ is Hopfian, $f_*$ is an isomorphism. qed
Corollary. If $M$ is a real or complex-hyperbolic manifold, then any degree 1 map $f: M\to M$ is a homotopy-equivalence.
Proof. Since $M$ is aspherical (its universal cover is the unit ball), and $f_*: \pi_1(M)\to \pi_1(M)$ is an isomorphism, $f$ is a homotopy-equivalence by Whitehead's theorem. qed
Remark. It is an open problem, first posed by Hopf himself, if any degree 1 map $M\to M$ of a closed oriented topological manifold is a homotopy equivalence, see here.
Corollary. In the above setting, assuming that real dimension of $M$ is at least 3, there are only finitely many automorphisms $\phi: H_*(M)\to H_*(M)$ induced by continuous self-maps $f: M\to M$.
Proof. Suppose that $\phi$ is such an automorphism. Then, by looking at the action on the top-dimensional homology, $f$ has to be a homotopy-equivalence. Mostow proved (Mostow rigidity theorem) that each homotopy-equivalence $f: M\to M$ is homotopic to an isometry. (Wikipedia article states this only for hyperbolic manifolds, but Mostow proved much more!) Lastly, the isometry group of a closed real or complex-hyperbolic manifold (of any dimension $>1$) is finite. This follows, for instance, from the Arzela-Ascoli's theorem and the fact that an isometry homotopic to the identity has to be the identity map. qed