Homomorphism
an operation is called homomorphic if it follows: $ψ(a⊕b)=ψ(a)⊕ψ(b)$
Endomorphism
An endomorphism is a homomorphism from an algebraic structure to itself. In other words, it is a map that preserves the structure’s operations and maps elements within the same structure.
Definition
Let $(G,⋅)$ be a group. An endomorphism of $G$ is a function $ϕ:G→G$ such that for all $a,b∈G,ϕ(a⋅b)=ϕ(a)⋅ϕ(b)$.
Similarly, for a ring $(R,+,⋅)$ an endomorphism is a function $ψ:R→R$ such that for all $a,b∈R,ψ(a+b)=ψ(a)+ψ(b)andψ(a⋅b)=ψ(a)⋅ψ(b)$.
In the context of vector spaces, an endomorphism is a linear transformation from a vector space to itself.
Examples

Identity Map: The identity map $id_{G}:G→G$ defined by $id_{G}(x)=x∀x∈G$ is an endomorphism.

Matrix Multiplication: For the ring of $n×n$ matrices $M_{n}(R)$, the map $ϕ:M_{n}(R)→M_{n}(R)$ defined by $ϕ(A)=BA$ for a fixed matrix $B$ is an endomorphism.
Isomorphism
An isomorphism is a bijective homomorphism. It is a map between two algebraic structures that preserves operations and has an inverse that is also a homomorphism.
Definition
Let $(G,⋅)$ and $(H,∗)$ be groups. An isomorphism from $G→H$ is a bijective function $ϕ:G→H$ such that $∀a,b∈G,ϕ(a⋅b)=ϕ(a)∗ϕ(b)$
Similarly, for rings $(R,+,⋅)$ and $(S,⊕,⊗)$, an isomorphism is a bijective function $ψ:R→S$ such that for all $a,b∈R,ψ(a+b)=ψ(a)⊕ψ(b)andψ(a⋅b)=ψ(a)⊗ψ(b)$
In the context of vector spaces, an isomorphism is a bijective linear transformation.
Examples

Vector Spaces: The map $ϕ:R_{2}→R_{2}$ defined by $ϕ(x,y)=(x+y,x−y)$ is an isomorphism between the vector spaces $R_{2}$ and itself.

Groups: The map $ϕ:Z→2Z$ defined by $ϕ(n)=2n$ is an isomorphism between the group of integers under addition $Z$ and the group of even integers under addition $2Z$.
Properties of Isomorphisms
 Bijectivity: An isomorphism is both injective (onetoone) and surjective (onto).
 Preservation of Structure: Isomorphisms preserve the algebraic operations, meaning that the structure of the original algebraic system is retained in the image.
 Inverse Map: The inverse of an isomorphism is also an isomorphism. If $ϕ:G→H$ is an isomorphism, then there exists a map $ϕ_{−1}:H→G$ such that $ϕ_{−1}∘ϕ=id_{G}$ and $ϕ∘ϕ_{−1}=id_{H}$.
Relation Between Endomorphisms and Isomorphisms
 Endomorphism: A map from a structure to itself that preserves the operations. Not necessarily bijective.
 Isomorphism: A bijective homomorphism between two structures that preserves the operations. An isomorphism between a structure and itself is called an automorphism.
Automorphisms
An automorphism is a special case of an isomorphism where the domain and codomain are the same.
Definition
An automorphism of a group $G$ is an isomorphism from $G$ to itself. Formally, $ϕ:G→G$ is an automorphism if it is a bijective homomorphism.
Examples
 Identity Map: The identity map $id_{G}$ is an automorphism of any group $G$.
 Complex Conjugation: The map $ϕ:C→C$ defined by $ϕ(z)=z$ (complex conjugation) is an automorphism of the field of complex numbers.
Automorphisms form a group under composition, known as the automorphism group of the structure.