Let $A$ be Abelian group of order $p^4$, where $p$ is a prime number, and which has a subgroup $N$ with order $p$ such that $A/N\approx\mathbb{Z}/p^3\mathbb{Z}$. Find all $A$ expect isomorphic.
1. **Understanding the problem**: We are given an Abelian group \( A \) of order \( p^4 \), where \( p \) is a prime number. We know that \( A \) has a subgroup \( N \) of order \( p \) such that \( A/N \cong \mathbb{Z}/p^3\mathbb{Z} \). We need to find all possible structures for \( A \) up to isomorphism.

2. **Using the structure theorem for finite Abelian groups**: According to the structure theorem for finite Abelian groups, any finite Abelian group of order \( p^4 \) can be expressed as a direct sum of cyclic groups whose orders are powers of \( p \). Therefore, the possible structures for \( A \) are:
   \[
   \mathbb{Z}/p^4\mathbb{Z}, \quad \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}, \quad \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z}, \quad \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}, \quad \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}
   \]

3. **Analyzing the given conditions**: We need to find which of these structures have a subgroup \( N \) of order \( p \) such that \( A/N \cong \mathbb{Z}/p^3\mathbb{Z} \).

4. **Checking each possible structure**:
   - For \( A = \mathbb{Z}/p^4\mathbb{Z} \):
     - \( A \) has a subgroup \( N \) of order \( p \) (e.g., \( N = \langle p^3 \rangle \)).
     - The quotient \( A/N \cong \mathbb{Z}/p^4\mathbb{Z} / \langle p^3 \rangle \cong \mathbb{Z}/p^3\mathbb{Z} \).
     - This satisfies the given condition.
   
   - For \( A = \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \):
     - \( A \) has a subgroup \( N \) of order \( p \) (e.g., \( N = \langle (0, 1) \rangle \)).
     - The quotient \( A/N \cong (\mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}) / \langle (0, 1) \rangle \cong \mathbb{Z}/p^3\mathbb{Z} \).
     - This satisfies the given condition.
   
   - For \( A = \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p^2\mathbb{Z} \):
     - Any subgroup of order \( p \) in this group would be of the form \( \langle (a, b) \rangle \) where \( a, b \in \mathbb{Z}/p^2\mathbb{Z} \).
     - The quotient \( A/N \) would not be isomorphic to \( \mathbb{Z}/p^3\mathbb{Z} \) because the orders of the elements in the quotient would not match.
     - This does not satisfy the given condition.
   
   - For \( A = \mathbb{Z}/p^2\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \):
     - Any subgroup of order \( p \) in this group would be of the form \( \langle (a, b, c) \rangle \) where \( a \in \mathbb{Z}/p^2\mathbb{Z} \) and \( b, c \in \mathbb{Z}/p\mathbb{Z} \).
     - The quotient \( A/N \) would not be isomorphic to \( \mathbb{Z}/p^3\mathbb{Z} \) because the orders of the elements in the quotient would not match.
     - This does not satisfy the given condition.
   
   - For \( A = \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \):
     - Any subgroup of order \( p \) in this group would be of the form \( \langle (a, b, c, d) \rangle \) where \( a, b, c, d \in \mathbb{Z}/p\mathbb{Z} \).
     - The quotient \( A/N \) would not be isomorphic to \( \mathbb{Z}/p^3\mathbb{Z} \) because the orders of the elements in the quotient would not match.
     - This does not satisfy the given condition.

5. **Conclusion**: The only structures for \( A \) that satisfy the given conditions are \( \mathbb{Z}/p^4\mathbb{Z} \) and \( \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \).

The final answer is \( \boxed{ \mathbb{Z}/p^4\mathbb{Z} } \) and \( \mathbb{Z}/p^3\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z} \).